That doesn't explain why people choose to have small families.

No, but that explains why that choice exists.

Not really. Humans have exercised control over family size for thousands of years via all sorts of different mechanisms. Modern birth control is certainly more convenient than the vast majority of ancient mechanisms, but it's not clear that the increase in convenience is why the modern world is a lot less excited than the ancient one about the command "Go forth and multiply."

Would you seriously argue that people choose to have children as a reasonably optimal selfish way of guaranteeing that they continue to have enough to eat once they're no longer capable of working?

Yes, seriously, I find nothing outlandish about this assertion. Why are you so surprised?

It's not a totally unreasonable argument, it's just very contrary to my personal experience. I don't see humans commonly engaging in a lot of decades off long-term thinking, and child-creating/child-rearing typically seems to be dominated by a lot of deep instinctual emotional cues. Parents often devote significant resources to caring for special needs children who are unlikely to grow into good providers. Parents seem to derive a lot of satisfaction and to compete for status based on the way that their children perform in school or child sports or other competitions, even though these are very weakly linked to any sort of economic productivity. Hutterite communities practice common ownership and thus have a more extensive social safety net than even modern societies, but traditionally have large average family sizes.

It's not obvious to me that humans reproduce for reasons substantially different from the reasons that other animals reproduce, and it's very obvious that most creatures aren't having children in order to secure their retirement. The tendency of parents to take exceptional risks in order to protect their offspring is also better explained by these traits being promoted by genetic self-interest than the idea that children are a rational agent's retirement plan. It seems profoundly weird to me that all the other animals reproduce because their genes tell them to, but humans just so happen to make the same exact decision for a completely different reason.

It's also not totally obvious to me that children are a particularly good investment from a long-term wealth or even a guaranteed income perspective. I feel like if most people directed the same amount of resources into securing their retirement via other means that they typically direct into child-rearing they would often end up better off. It's an interesting thesis though, and it has some neat fits to the data. The falling birthrate, it could be argued, is due to the fact that there are more long-term investment opportunities now than there were in the past and people have chosen to diversify their investments. At the same time, modern decreases in child-mortality and death in child-birth might make children a more attractive investment than they were in the past, so there are definitely some issues.

Because if you're managing the number of your children, you're managing the number of children who'll grow up to adulthood.

But why choose to have fewer children grow to adulthood rather than more? If my children are more likely to survive to adulthood then I should have more children, not fewer, since child birthing and raising is now a better investment than it was before.

Clearly, people are interested in more than that and on a very regular basis choose NOT to maximize the spread of their genes.

But why is that even an option? What evolutionary advantage is so adaptive that even though it leads to obviously maladaptive behavior like choosing not to have children that you could easily support the adaptation is still a net-positive on average?

I'm not sure what the difference is between [Women have attractive alternatives to just being a mother] and "achievements other than sex and children have become markers of status".

The difference is you're talking solely about status and I'm talking about a much wider context.

I think the more fundamental difference may actually be that I'm talking about population level processes and you're thinking about things on the level of individual decision makers. I think most human decisions are constrained fairly tightly by the culture that they find themselves in and that copying high status individuals is one of the driving forces of cultural change. I think cultures that have family size as a primary marker of achievement and status don't create the opportunity for people to support themselves by going to graduate school to study science or math instead of raising a family. Social status is very much about having a place in the community, and cultures don't create good attractive places in the community for occupations with low social status. An individual may choose to pursue a Ph.D. in math rather than a more conventionally high status career in finance, but a society that doesn't see educational attainment as a mark of status isn't going to have a career track for graduate students at all.