I would agree with this statement. I'm no musicologist, and my knowledge of music history has some gaping holes in it, but I would go so far as to say that there hadn't been such a dramatic advancement (not as in 'better'; just move toward modernism) or shift in styles or thought or attitudes toward music as with Schoenberg's twelve-tone technique. Hauer should get some credit here for his twelve-tone ideas that were ultimately overshadowed by Schoenberg's system. One could maybe argue that it was an idea waiting to happen...
In a video on YouTube about Schoenberg's (wonderful) piano concerto, Mitsuko Uchida talks about how Debussy's rebellion of traditional harmony was to kind of chuck it out the window and leave it behind, while Schoenberg grabbed it with both hands and actively fought against it. He was a big deal, and other people of (roughly) the time (like Stravinsky, Mahler, Scriabin) have kind of lost their shock-value or controversy, in my opinion. While they're not universally admired, their works hold solid places in the repertoire. It's been 100 years and Schoenberg is still a point of contention for many people, I'd say.
After Schoenberg, the next BIG one for me would be Milton Babbitt... I'm surprised he didn't make the list already, and I have lately been obsessed with his work. His Three Compositions for Piano, as I recall, was the first piece to use "total serialism" or whatever you want to call it, apparently coming in a few years before Frenchmen like Messiaen or Boulez (rightfully on the list). I have come to be really fascinated by his stuff. I can't say I understand it or get it, but his string quartets (the second and sixth in particular), clarinet quintet, and even his first piano concerto are at the very least extremely interesting if not downright enjoyable.
But he still seems to garner quite a lot of spleen from music critics and musicians who see (hear) him as nothing but a bunch of noise. The same I suppose could be said for Boulez and many others who worked with serial techniques; I just picked him because he was (by some accounts) the earliest adopter of the idea. I still find Boulez's works hard to warm up to. Babbitt far more approachable, to me.
Aside from those already mentioned, I'd say Ferneyhough and Finnissey are good contenders; I find their works wholly unintelligible, but then again... haven't given them much effort. Same goes for Stockhausen, Reich, Berio, Schnittke.
Great topic, by the way.