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Some people will want to have large families, but on average it seems that most people don't.

If the tendency to have large families is heritable in some measurable way, maybe because of beliefs or personality traits, wouldn't the carriers of whatever makes them have more babies eventually overtake the rest of the population?

It may not be the case that this is how it will work, but it's not a foregone conclusion that "on average" most people will continue to not want to have kids even in the "first world."

(Don't Mormons for instance have a fertility rate well above replacement despite also having higher than average incomes?)

That seems plausible. I think it's reasonable to suppose that people who grew up in large families might themselves prefer to have large families, but I also suspect that personal preferences like that aren't all that sticky across more than a few generations. Maybe it balances out. For now, we can just say that most people in well-off societies aren't opting to have large families and that doesn't appear likely to change in the near future.

> I also suspect that personal preferences like that aren't all that sticky across more than a few generations

There is pretty strong evidence that the variance in the number of offspring is about 50% genetic.

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