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. Everybody knows this.

In a sexual species, an individual organism doesn't evolve; it keeps whatever genes it's born with. An individual is a once-off collection of genes that will never reappear; how can you select on that?

You select on it because the variation in genes this individual winds up with makes it more likely to survive and reproduce. The organisms aren't competing to acquire better genes, they're competing to survive and reproduce. You select on the organism's genes by whether they actually do survive and reproduce more than the ones with/without certain traits.

Your rationalization that the mice are evolving to extinction is flawed. It contradicts this statement you made:

The rarer males become, the more reproductively valuable they become - not to the group, but to the individual parent.

If the mice introduce a "males only" gene, other mice will start introducing a "females only" gene to compete. 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. They compete, and your females get to reproduce with the strongest of the males. Everything moves back into balance.