Having a man to help raise a woman's children greatly increases their prospects

This is no longer true in countries where women can earn their own living.

So, women would prefer [...]

Don't generalize. Women are individuals, not a monolithic bloc.

Men don't what to raise children that aren't theirs.

Again, don't generalize. I'd much rather adopt an orphan than increase the amount of living people, all the while knowing my genes would much rather be copied. I don't fear my moral inclinations will be outbred; I can just teach them to my adopted children. Every man (and woman, and genderless person, and all variations thereupon) has hir own motivations and interests. Ignoring individual differences is one of the problems with this particular brand of sexism.