No, evolution does not do that. But this is a hard question to answer succinctly, because of the layers of concepts involved. I'll have a try.
First, consider one of the core notions in Buddhism: that our normal view of reality suffers from three illusory aspects. We don't normally see that the thing we usually call "real" -- tables, chairs, electrons, our brains, the evolving of species of organisms through natural selection, etc -- are impermanent, do not satisfy our desire to be free of suffering, and do not comprise that entity we call "self". A major point of Buddhism is to get to understand and experience the "true" nature of reality.
OK, so far so enigmatic. The point is, according to Buddhism what we see (feel, smell, etc) "out there" is not "real" -- it is merely the way whatever it is that is real appears to us. A well-used but very good analogy is the movie The Matrix. We are, in a very true way, in a Matrix. There actually is a "reality", but what we experience is not It. What we experience is just how It appears. And that idea isn't unique to Buddhism or even to religion.
Now within that set of experiences -- the things inside our Matrix -- are matter, energy, time, and so on; in other words, all the things of which a materialist scientist would say "that's all there is to reality". But in Buddhism, the insight is, no, that's not "all there is".
So to your specific question. The reason evolution -- and 200,000 years of the homo genus -- doesn't disprove kamma is because evolution is a concept inside the Matrix. Kamma, on the other hand, sits outside it.
Here's another analogy. Imagine you're a player of an online video game, like Call of Duty. Within each specific match, you can increase your score, gain points, get better weapons. Those are all like evolution. They happen inside one specific match. But you also have an overall ranking, or "experience points score" (I forget if that's what CoD actually calls it, but the idea is accurate) that lasts from game to game. That's like kamma. So asking if evolution disproves kamma is like asking if the fact that the current match has only been running for 5 minutes proves that no one can have a total "experience point score" of 10,000. The answer is, no, of course not. One is a concept within a game-match/samsaric-existence-instance; the other is a concept outside or "across" game-matches/samsaric-existence-instances.
One last way of thinking. Modern physics allows the possibility of multiple universes within a multiverse. So our current universe's 200,000 years worth of humans-able-to-attain-nibbana may be only one of billions of examples of such instances of human species across the multiverse, and maybe rebirth happens across universes.
So, again, no. The fact that we have a localized phenomenon called Darwinian evolution (which I accept, as far as it goes) really says nothing about the "global" (all-of-reality-scale) nature of things.
(See, I told you I'd find it hard to be succinct!)