That is probably the best argument against it, but I still think it is a weak argument.

Countries with populations that live longer tend to have fewer children and care for them better. Look at Europe and even he United States, the average amount of children per pair of parents is 1.5 to 2.5 depending on where you look. If you pick poorer places that have way more kids.

I think its clear the mechanism at work here is wealth and not age directly. Richer countries make richer people and richer people live longer. Then something happens and they have fewer kids.

For many social and health issues it seems possible to go the either way. Some place increase wealth to get health, others increase health and get wealth. If someone poor today could be made to live until they chose to die, would the knowledge they accumulate in their unnaturally long life lead to increased wealth in their community? I think it would and I think it would lead to the same long term reduction having too many kids.

Or you could summarize it as: wealth and health are bidirectionally causal. In any given situation, one of them will be the limiting factor to increases in both.

If we can control our population growth that isn't inevitable. If we can't we'll get that with or without aging.