I think you're misunderstanding an important part.

Reproduction is made conditional on some quantum event, so that it happens in some futures and not in others - OK.

I think the idea is that you flip a quantum coin, and if it's heads you do nothing, otherwise you replace yourself with your child. So the population of each branch stays the same.

But the coherent superposition part doesn't make sense to me either. And the scheme also doesn't say what prevents people from reproducing the normal way and driving up the per-branch population.