I'll take a stab at this question, which is quite complicated to answer. The simple (but unsatisfying) answer is that songs with these chords are popular because that's what we're used to hearing in popular music today. There's no necessary acoustical or theoretical reason why these chords must be "superior" to all others. Many other musical systems existed historically and continue to exist today, which produced music that was immensely popular in various cultures without centering on these chords (or not centering on chords at all).
The slightly more nuanced answer is that once you accept the major diatonic scale with triads built on each degree as a theoretical system (again, something that's a product of complex historical and cultural forces), these four chords -- out of seven possible -- give you the most variety of progressions that incorporate both traditional chord motions and allow novel permutations.
That's the TL;DR version. Below are further details.
The Danger of Circular Reasoning
It's easy to fall into the temptation to try to justify one's current culture and preferences by appeals to some sort of external logic. Ultimately, however, this can lead to a lot of circular reasoning. For example, some answers postulate that I, IV, and V are the best "central" chords because they cover all notes of the major scale within them. While that seems a reasonable criterion for harmonization possibilities, it doesn't explain why, say, I, ii, and V couldn't be used as an alternative system, as they have the same characteristic. (And there are other examples of chords that cover the scale.) Similarly, one could note things like ratios in the harmonic series and postulate some connection between the fifths and fourths that show up there and the connections between fifth/fourth root motion in triads. But again, there's no clear connection -- psychoacoustical research does show that the structure of the harmonic series can lead to consonance and dissonance of harmonic intervals, which is one reason for the existence of triads in modern pop music. But why should that have any connection to consecutive intervals between chords that aren't sounded together? And why would only the intervals between the roots be important?
I'm not saying there can't be some truth in some of these explanations, but a lot of them are taking a system we already have in place and creating an ad hoc justification after the fact.
I/IV/V Was Not Universal (nor was the ubiquity of diatonicism or major mode)
The first thing we must accept is that Western pop music is only one possible musical system that developed through a lot of historical and cultural processes over centuries. After that, we can begin to take apart the reasons why it developed the way it did, and why things like the I/IV/V set of chords became so common.
There are lots of other assumptions we need to work under here, like the preexistence of something resembling the seven-note diatonic scale (developed in ancient Greece and gradually transmitted over the millennia in various forms), the usefulness of triads (a concept only developed in the early 17th century), etc.
But let's pick up the story around the year 1700, when popular music ("folk music") of Europe sounded a lot different from what it does today. Composers of the 1600s would be just as likely to pick I/ii/V as a central group of three triads in a major mode, rather than I/IV/V. The reasons for this are complicated (and would take us down even more digressions), but it's important to realize that I/IV/V was a relatively late development in Western music.
As triadic theories developed in the 1600s, I and V were obviously going to be central, as they had framed the structure of cadences since they began to resemble modern form in the mid-1400s. But the next-most-important chord was not very clear. The role of ii and ii7 in cadences before 1700 was more prominent than IV, and the more modal character of folk music of that time made more heavy use of minor options.
That's another assumption in the question -- why is major mode dominant? Why do we analyze the chords as I/IV/V/vi and not i/III/VI/VII? Indeed, many modern pop songs emphasize the one minor chord as more of a tonic/home, so that ambiguity is present in pop music. But the emphasis on the major mode as a theoretical center didn't begin until the 1700s. Today, we tend to think of the minor mode as "inferior" somehow or at least denoting some negative emotions (sadness, distress, etc.), whereas historically before the early-mid 1700s this was not true. And the minor mode developed lots of other non-diatonic chord options, which resulted in a richer harmonic palette. Yet the theoretical simplicity of the major mode (along with the newly formulated arguments on the basis of supposed connections with the harmonic series) placed major at the center of a lot of music theory by the mid-1700s.
Another assumption brought up in the previous paragraph is that of diatonicism. Today, oodles of popular theory books start with an assumption that chords are built on diatonic scales, and harmony is therefore essentially diatonic. Many college courses start with a semester or even a full year focused on diatonic harmony. But this is an anachronism when used to discuss classical music, which was definitely not diatonic, even in its simplest conception. Even little 5-year-old Mozart knew to put a #4 leading to 5 on his half cadence in the middle of his minuets. So-called "chromatic harmony" was not a more advanced element tacked onto a diatonic system in baroque and classical music. Instead, chromaticism was fundamentally "baked in" to even the simplest exercises taught to beginning students. Our "diatonic" harmonic system is a sort of myth, created by later pedagogues -- prior to 1800 or so, monophonic chant was founded on diatonic scales, but polyphonic harmony was decidedly not.
The Historical Emergence of I/IV/V as Central Triads
So when did all of this change? A lot of it came in with the theories of Rameau in the mid-1700s. He emphasized the primacy of analyzing music as built on triads. In his initial theories, he didn't foreground IV, but his later theory finally builds the I/IV/V structure as central to creating harmony in a major key. (This comes out of a sort of mystical obsession with ratios, which frequently ignored or reinterpreted actual musical practice to fit his theories.) He uses the harmonic series to create an ad hoc justification for the major scale and triads. Later in the 19th century, Germanic Stufentheorie (scale-step theory) builds this concept of diatonic triads on each step of the scale, which allows the creation of a diatonic world, as discussed in elementary theory textbooks today.
One could write an entire book on why I/IV/V gradually came to be central chords, but this outlines a bit of it. And then this new system began to spread in popular music. In various countries, it happened on different timelines and in different ways. In the U.S. (most relevant to global pop), my guess is that it was mostly promulgated in the 19th century through the harmonic vocabulary of sheet music pop songs (think Stephen Foster) and revivalist hymnals (where the hymns were composed to make congregational singing very easy, and the composers were not necessarily versed in complex harmonies). These trends supplanted the earlier modal styles (often infused with more minor mode) and systems with a more varied harmonic vocabulary. One can compare ballads written in the mid-1800s with those in the mid-1600s to see how radically the I/IV/V major-mode theoretical chord system changed popular music. Pockets of older systems survived in isolated communities, for example, the shape-note singing of the American South which preserved hymnody with more varied harmony and texture (and which still contains some practices closer to 17th-century hymnody).
I've skipped over a lot here, but one other important element to bring up is also the role of Spanish guitar in cementing a chordal view of music beginning around the 17th century. (This followed a rapid dissemination of guitars around Europe, as an alternative plucked string instrument that was louder, fun to play, and perceived to be easier than the lute.) Early alfabeto notation used predecessors of modern chord symbols, which may have influenced the increasing trends toward chord classification and eventually Roman numeral development in the early 19th century.
With the minor mode, chromaticism, and a variety of harmonies increasingly abandoned in favor the major-mode diatonic system that borrowed the theories of Rameau (and others who build on such theories) to emphasize I/IV/V, the stage was set to gradually take over a lot of popular forms in the 19th-century as sheet music and hymnals spread these structures to the masses.
(Note: I should qualify all of this by saying that this wasn't just caused by Rameau and his followers promoting this new theory. I've necessarily oversimplified things. Rameau was a product of his era, and there was already popular music using more major mode and using I/IV/V developing alongside these theoretical systems. But those are harder to trace than treatises that systematize such developments.)
What about vi?
So far, I've neglected vi. It was a chord that had been used for many centuries in "deceptive cadence" motion, so it's an obvious choice for some variety. But why settle specifically on I/IV/V/vi as the vocabulary for modern pop?
The proceeding discussion was necessary to establish that the concept of a major diatonic scale with chords built on each scale degree is NOT the only possible musical system or the most "natural" one -- it was a very gradual historical development. But once we have that major diatonic scale with triadic harmony, why choose these particular chords?
Let's look at the alternatives. The vii° chord has functioned since the 17th century primarily as an incomplete V chord. Since we need V to make the same sort of cadences we've been making in Western music for 500+ years, vii° gets us nothing significant more. That's aside from the fact that it is diminished and therefore branded in modern music theory as somehow deficient compared to the major and minor triads.
Then we could look at iii. But the role of iii started to be downplayed in the major mode since before the concept of the triad even emerged. Studies have shown that it decreases in usage in the major mode even in the early 1500s. It was just never a significant part of the standard harmonic vocabulary that developed for the major mode. (Mozart could -- and did -- write an entire symphony in major without even using a iii chord.) Aside from certain uses in long sequences, it was just never that common.
So the only other chord left is ii. As noted above, ii was actually a more common option than IV in earlier styles as a predominant, but Rameau undermined that in favor of IV. (Subsequent theorists have found various other ways to justify ignoring ii in favor of IV, frequently invoking abstract principles that had nothing to do with historical musical practice. Nevertheless, IV gradually caught on.) Restoring ii does give us a ii-V-I turnaround, which is very popular in classical, jazz, etc. styles. And it is an interesting "more minor" predominant that could color the harmony. But there's not a lot going for ii when you already have IV. One could imagine an alternative universe without Rameau where we used the four-chord group I/ii/V/vi instead, which makes for a nice circle-of-fifths vi-ii-V-I sequence. It would be a more minor world, and one that sounds more "modal" than the way that music developed over the 18th and 19th centuries.
Anyhow, if we accept the I/IV/V axis for the major mode to revolve around, vi is the most logical next chord. Aside from deceptive cadences, it gives a hint at relative minor as well as a "minor turn" in general to contrast with the other major harmonies.
In pop music in particular, I also think its inclusion partly comes from the prominence of the descending thirds sequence (I-vi-IV) over the circle-of-fifths sequence (vi-ii-V-I). Everything from doo-wop and bubblegum pop to various ballads in the 1950s began to center around the traditional thirds sequence cycle. I'm not sure why this became so popular, though the fact that there are more common notes held in the descending third sequence was always a reason for its popularity (and gives it flexibility for melodic harmonizations). You can't play a lot of those songs without vi, though you can often find substitutions for ii when it occurs (either IV or vi).
The Staying Power of I/IV/V/vi
Hence, we arrive at a four-chord I/IV/V/vi minimal vocabulary. I think it has had staying power in pop music because of its variety. You have a minor chord for contrast, as well as the deceptive cadence (V-vi) and descending thirds pattern (I-vi-IV). You have a nice strong predominant (IV). You have the modal ambiguity I mentioned earlier, where vi-V-I can easily be reinterpreted as the move to relative major (i-VII-III) that shows up as a standard musical pattern since the 17th century. You have a nice mixture of possible root motions: fifths/fourths (V-I and I-IV), thirds (I-vi-IV), and seconds (IV-V-vi), which adds to the feeling of harmonic flexibility with a minimal chord set. And, as we've seen with the development of patterns like the Axis of Awesome progression (I-V-vi-IV) or the "sensitive female chord progression" (vi-IV-I-V), the various permutations can yield more than just standard authentic cadence (IV-V-I) and 50s pop (I-vi-IV-V) progressions.
You can't get this amount of variety by using another subset involving ii. You'd either end up with a more symmetrical but more boring I/ii/V/vi or a pattern less suited to pop progressions and duplicating the predominant function (I/ii/IV/V). As mentioned, the other diatonic triads (iii and vi°) were just never real contenders given their lesser importance historically.
So, through process of elimination, once you've settled on the idea that music should use the simplified major diatonic system with triads (the assumption in pop theory for the past century or so), the I/IV/V/vi system just gives greatest "bang for your buck" in terms of variety and flexibility, if you're not willing to learn or use more chords.
