John Baez has had an interesting series of posts about this on G+, but IIRC the gist of it is that even though both Grigori Perelman and Yitang Zhang were somewhat outside the mainstream of academia, their work was not. Both of them built on well-known theories and methods to prove their groundbreaking theorems. Perelman's primary tool, for example, was Ricci flows, which had been used at least 25 years before his work. (I know even less about Zhang's work)
I think the mathematical edifice is sound, but I can certainly see how being as divorced as possible from typical faculty responsibilities makes it easier to do original / groundbreaking research. Too many professors I know spend the bulk of their time attending to e-mail, grant writing, and traveling to give talks about work others have done. What little remaining time they have is often devoted to teaching. Some of them openly complain about how little time they have to actually think about things, while others seems to roll with it (being famous in your field is not a bad gig). Worse, the more successful you become the more these tertiary responsibilities encroach on your time. It's an open secret that the best-known professors in certain fields basically sign their names on the papers their postdocs and grad students write and that is more or less the extent of their research contribution. Cutting all this out, even if it means moving in with your mom in Moscow, could indeed be a viable option...
The bare truth is this kind of mathematics (and, say, theoretical physics) is "hard" and requires a kit of thinking and thus a lot of time. Breakthroughs are named that for a reason -- they involve a distant, non-local solution. If you're doing your research on a topic you will most likely only focus on local improvements to the problems, simply because they lead to more steady results -- they are both safer and don't require you to pretty much think all the time about the problem at hand -- you can't really expect every mathematician to prefer that kind of work.
What I'm trying to get at is that there's two ways you can think of academia: 1) roughly the set of people in traditional institutions, and 2) a basically unorganized set of people that happen to come together through working on related problems, in related ways.
2) is arguably the purer view of academia, and in that sense, it's as strong as ever.
I agree with you that there's deep, serious problems with 1), and they have to do with Pournelle's law, (in the US) with Bayh-Dole, and a litany of other hard, nasty problems of social life. But, even in that case: is that any worse than other cultures?
But Zhang was teaching as a adjunct (or at least, a non-tenure-track instructor) before he made this breakthrough and skyrocketed (apparently) to full professor. Adjuncts generally have enormously less time for research than full time professors do (at least if they want to make a borderline living wage).
