Triads are the basic building blocks of classic harmony, but in Jazz the basic building blocks of harmony are tetrads (7th chords). Why does Jazz use 7th chords as the basis of their harmony instead of triads?
I absolutely see where your question is coming from, but I think there are a couple of misconceptions buried in there that should be rooted out first:
1) This is just a terminology thing, but it's helpful to be clear that "triad" doesn't only mean a three-note chord (post-tonal theorists tend to use "trichord" for that more general structure), it's a three-note chord in which the chords are built out of stacked major and minor thirds. Hence, there are only four possible triads: 2 Major 3rds=Augmented, 2 minor 3rds=diminished, M3-m3 = Major and m3-M3 = minor. Both jazz and "classical" make use of all of these, although jazz makes far greater use of augmented triads. (BTW, I haven't seen the term "tetrad" before, I don't think it's in general use.)
2) In light of that, seventh chords aren't some other kind of harmony, they are just what happens when you stack one more major or minor third on top of a triad. There are 15 possibilities here (there would be 16, but M3-M3-M3 doesn't generate a useful seventh chord for reasons not worth going into here). This is the first big difference between common-practice "classical" music and jazz, because, for the most part, common-practice music only uses five of these possibilities (m7,M7,dom7,dim7,half-dim7), whereas jazz uses all of these plus most if not all of the others pretty regularly.
3) The biggest difference between the two styles, however, is that harmonies with even more stacked thirds—such as 9th, 11th and 13th chords (15th and above are pointless for reasons not worth going into here)—are quite common in jazz, whereas they are only just starting to be used in the late-Romantic era of common-practice music.
So I would say that the basis of both kinds of music—and really the vast majority of all "western" music— is the interval of the 3rd, both minor and Major. More and more 3rds can be stacked, and, after the triad, each new 3rd added to the stack increases the dissonance of the harmony.
So it isn't that jazz uses 7th-chords as its basis, it's that jazz uses 7th-chords FAR more freely than common-practice tonality did, and it uses 3rd-stack extensions beyond 7th-chords all the time as well. "Why?" is a difficult question to answer; to some extent the answer is just "because jazz composers like it." But the increase in tension caused by far freer use of dissonant harmonies is definitely part of the reason. As audiences became more-and-more inured to the basic dissonances of the common-practice era, composers started using that dissonance more freely and started adding newer dissonances in order to keep the tension high, and jazz composers are no exception to that.
EDIT TO ADD: But as Roland Bouman points out in his comment, there very quickly comes a point when dissonance like this just becomes color in its own right, not necessarily creating tension anymore. Once a listener no longer expects any particular resolution, the intervals can just "be themselves" if you will. See also other post-tonal music.
I personally don't think of "the basis" of jazz harmony being all that distinct from more traditional triadic harmony. vi-ii-V-I is a perfectly valid progression in traditional harmony, as it is in jazz.
I tend to think about it modally: A chord notated as "C" generally indicates a chord derived of the C major scale, meaning it could be a C triad, a C6, a Cmaj7, a Cmaj9, a C69, and so on (generally omitting the 4th/11th, as it's a bit dissonant for a tonic, resolved chord). One might also interpret it as coming from the C Lydian mode, indicating any of the above chords or with the addition of the #11. Which of these is played in any given situation will vary, depending on the preferences of the performer and the style being played (swing vs. bebop vs. post-bop vs. free jazz).
It's not that the "basis" of the harmony is something other than a triad, but that the stylistic framework allows you the freedom to use as many or as few extensions to that original triad as you wish.
It's not 7ths instead. It's 7the as well. And 9ths. And 11ths. And altereds. All this because using triads alone gives very limited scope. O.K. it worked for a lot of the classical and baroque composers, but things move on, and through pushing boundaries it has been found that extra notes give extra harmonies. What a surprise ! But back in the early days, triads were obviously enough...
Including the 7th in the chord (or playing it instead of the 5th) lets you hear a clear difference in the dominant chord (major 3rd, minor 7th). In the V-I chord progressions it's particularly useful because you hear the resolution from the tritone formed by the major 3rd+minor 7th to the root+major 3rd of the tonic.