I would've remembered that women are already born with a finite amount of eggs.

Isn't that currently in doubt? I recall that in other mammals marrow stem cells have been shown to differentiate into follicles, but no studies have shown this to occur in humans yet.

Weak rebuttal: Another reason people don't have children in their 50s-70s is that chasing kids around is tiring. If women knew that they would at least still be young physically at those ages, they might think ahead and have some eggs frozen.

Or have eggs generated through biomedicine practices of the age. We're pretty close to differentiating sperm cells already, eggs can't be that far off.