Schoenberg isn't music, it's numerology.
That's why it's not popular.
Beethoven endures because Beethoven wrote good music that was rooted in the tradition.
The common practice period was an outgrowth of what came before; not some totally new invention that defied all that came before it like serialism and all other manner of modernism do.
The entire point of Shoenberg, like all modernism and atonalism, is to reject hierarchy in favor of "equality", generally as a political statement. However, there is no good art without discrimination and hierarchy. For example, dissonance is only beautiful within a hierarchy of consonance.
"Music" like Shoenberg differs from Beethoven on this crucial axiom: The latter's music is born of techniques rooted directly in the listening experience, whereas the former's is rooted in subversive, pseudo-intellectualism. Such systems, which are present in all atonalism, are ultimately a dead end.
You say for example that Shoenberg's music is "a thing to be dissected". This has never been the point of art and music throughout human history until the 20th-Century subversives like Kandinsky came along to mask their low ability. The point of creating art or music is an attempt to rival the beauty of nature herself. The Fjords of Norway and Hohenzollern Castle are alike in that they have stood for ages, and their beauty is still revered; it is self-evident across time. There is no need to "dissect" or "explain" what it all "means'. To gaze upon it beauty, and the mastery of craft it took to create it, is an uplifting experience in and of itself.
These abstract conceptualists who insist that art and music are actually about or at least better when we can play some trite game to figure out what the artist is "saying" — which, as a funny note: These people say music is "subjective" in quality, but apparently it can convey the artist's intent objectively despite this — and that the "meaning" is what it's all really about, don't seem to realize that they could get their message across a whole let more effectively and clearly if they just wrote it down.