You can't "evolve to extinction." Evolution does not "operate on" organisms. Evolution is created by organisms competing against individuals of their own species to survive and breed. Individuals with valuable traits are more likely to survive and reproduce. Then their offspring carry the trait.

Aaand sometimes this results in extinction. I don't see how any of this relates to your first sentence.

Also note that "valuable" traits are defined by reproductive success, making the ultimate player in this game the allele, as Eliezer said. This doesn't contradict your view of evolution as being caused by individual competition: differences in allele frequency are caused by the interaction of allele-bearing organisms.

The organisms aren't competing to acquire better genes

This was never suggested or implied. It was specifically noted individuals can't acquire new genes themselves, IIRC.

If the mice introduce a "males only" gene, other mice will start introducing a "females only" gene to compete.

No, they won't. They could just as easily have no more mutations and die out first.

You may argue that males are better since they can impregnate more females, but if there are almost no females left, and you start birthing females, suddenly all of the males are competing to mate with your female children.

That's totally correct - if lack of females becomes a problem, birthing only females could be a good idea. But there's nothing guaranteeing such an event will occur.