In any type of harmonic analysis the bass plays a big part in determining the function of the harmony. There are even special ways to describe what note of the chord is in the bass (known as inversions) and notation for chord symbols to make sure the right note is in the bass (known as slash notation).
So why does the bass note play such a big role in harmony?
There is a rather more fundamental, physical reason for this than so far mentioned: the bass fills not only the bass frequency range, but its harmonics actually reach well into the midrange where all other voices have their fundamentals! In fact, since the bass has typically the strongest amplitude1 of all tuned instruments (save perhaps trumpets, lead vocals etc.), the middle voices have a hard time to really isolate themselves from these harmonics, at any time. Their harmonic function is thus always determined to a large degree by what interval they play over the bass.
1 This has again a particular reason: to make up for Fletcher-Munson, so the bass won't seem much quieter than everything else.
This is not a rigorous, scholarly answer, but a useful one: There is a simple, general principle in writing Western music that has been mentioned by many people over the centuries. It basically says that a piece of music has two important components: the melody up top, and the bass line down low. The chords are determined by filling in the spaces between the melody and the bass line to create musical effects that are pleasing. No matter how pieces of music are written, if you look at the music after it's written, you can see the sense of this.
When students study Western music theory, or "common-practice-period tonal harmony" as it is officially referred to, in the style of J. S. Bach, a frequent homework assignment they encounter is to take a short melody, then construct a bass line beneath it, then to write notes in-between to create chords to fill in the arrangement, according to the academic rules of part-writing. This is certainly not the only method of writing music, but it is fair to say that this method has been extremely common in all styles of music ever since the time of Bach.
A lot of Western music falls into the broad category of primary melody on top, a lot of stuff in the middle, and bass line on bottom. The outside lines--melody and bass--are easiest for our ears to distinguish because they are the lowest and highest boundaries of each vertical slice of harmony. The inner voices are often (but by no means always) less important as melodic lines than they are for determining chord quality--the moment-to-moment color of a piece.
Therefore, if you know the bass, harmony, and melody of a piece of music, you know a lot about it. You know the shape and color of each vertical slice, as well as the contours of the most prominent horizontal lines. In fact, in many styles, there are idiomatic "good" ways to realize inner voices given a melody, bass and harmony, whether it be the figured bass notation of the Baroque or much freer jazz and pop styles.
Therefore, analyzing in terms of melody line, bass line, and harmony captures the "essential" identity of many types of music, from Baroque arias and dances to pop chord charts. An experienced performer can satisfactorily replicate the "fundamental" music from these sources.
I want to start my answer by saying: I don't know. This is a very profound question. It does not have a trivial answer. I don't think anyone knows the answer for certain.
Because your question isn't just about how Common-Practice Era Western Music Theory Type analysis proceeds. It's about how we think about what a chord is. What is it about how we hear chords that makes us say that CEG is in an important sense the "same thing" as EGC? Why do we conceptualize the various inversions of a chord as, er, "a chord"?
So this is a hypothesis: we think about music as we hear it, and individual pitches rest on a fundamental pitch because of how physics work, so perhaps we bring the same thinking to harmony. I hypothesize our thinking about harmony proceeds from the assumption that harmony is really multiple voices making a single pitch with complicated overtones.
When you hear "EGC" (the first inversion of C major), even if produced by three separate instruments, maybe you hear it as if it were a single voice singing a single pitch with those overtones -- but we know intuitively that in nature that cannot be: no single resonant pitch can have the structure of EGC, because overtones simply don't happen in that order, so it implies to us an unheard lower C supporting that chord.
That's my guess.
The bass is so important because it imposes its presence on all the other sounding notes via three main characteristics:
In a visceral sense, the bass is the below the other tones in the music, so it has the perceptual role of supporting them. An analogy can be made to the foundation of the house, which supports the rest of the structure. You don't start building a house until a good foundation is laid down, and the foundation determines how the rest of the house is built.
