There have always been nonwhite people as qualified as white people, and there have always been women as qualified as men, but it wasn't that long ago that corporations, governments, and other institutions routinely excluded those people or relegated them to menial work like cleaning floors. Discrimination based on genetics doesn't make sense, but it happens nonetheless.
After all, there was a time when European scientists were convinced that whites were a genetically superior breed and all other races were genetically inferior. They had elaborate rationalizations for why it was simply the natural and necessary order of things that whites remain dominant, because they were intrinsically more intelligent, competent, rational, and moral than the "degenerate" breeds. That belief had no actual basis in fact, it was irrational and stupid, but it was taken for granted by Western civilization for many generations. So it's not really surprising that in a future like Gattaca's where there's actually a basis for the notion of some people being genetically superior to others, old habits of thought might reassert themselves. It's just that in that future, the definitions of the "superior" and "inferior" breeds are based on something other than skin complexion. But it's still essentially racial discrimination.
Not quite.
See, when they (grantedly illegally) sample Vincent Freeman's DNA, they aren't going to find out solely that he was conceived naturally. They're also going to find out that he has the genes to be really, really, really, really smart and really, really, really, really disciplined (really, as close to pathologically disciplined as you could ever want).
They'll also find out he has a heart condition, sure. So yeah, he's going to be grounded from flight duty, but why in the world would they turn away someone with a naturally occurring grade-A genotype in every other respect, just because it's naturally occurring? It's like if someone handed you a diamond, better than most of the diamonds you usually take, but you throw it out because it has a minor imperfection that has nothing to do with the task you were going to assign to it, and because it had formed in the Earth's crust instead of in a diamond plant.
It's not enough to say that his physical infirmity would have kept him from being part of Gattaca, or at least part of the gene-blessed caste. Irene Cassini also has a congenital heart condition, and she works at Gattaca openly!
It makes me wonder how biased Vincent is in his narration of this "dystopian" gene-biased future. Maybe he could have gotten a job at Gattaca, but knew he could never go to Saturn with his heart. And for good reason--sending an astronaut who has an even chance of keeling over dead halfway to Saturn is a ridiculous chance to take, that cannot be supported by a rational decisionmaker.
So maybe Gattaca's society isn't really so bad, and Vincent isn't really so noble. Maybe the entire charade was motivated by more than a desire to just be successful. Maybe Vincent had to be the hero of his own personal drama, selfishly and wrongly risking the success of the Saturn mission, and even the lives of his comrades, because of his own pursuit of glory.