[Epistemological Status: I remember hearing all of these things but I could be wrong about a bunch of them. Take this more as my intuition than as a fact]
Right now, the age distribution of a given country is considered to be a very important factor in its economic development. This is because
- Elderly people are less likely to make major (retail) purchases before they've already accumulated the things they neeed over the course of their lives
- Elderly people are less likely to engage in professions because
- They are more likely to have enough wealth that working is unnecessary
- One's age often inhibits professional capabilities
- Elderly people are more likely to require familial support from their children and grandchildren; putting strain on their ability to work or their level of productivity while working
A lot of the economic strains noted above correspond to either the effects of aging or the economic situation of the elderly. Of the two of these, I think the effects of aging are much more important. While aging is a biological reality, things like the desire to continue working after being alive for a long time can change with culture (and even now, I know many elderly people who would like to continue working).
For this reason, I think anti-aging treatments targeting effects of aging rather than life extension are likely to be an overall economic benefit. It will give the elderly who want to work but can't because of health reasons the opportunity to do so. The result may be a larger workforce and higher consumption (even if that consumption is focused around getting toys for the grandkids).
Similarly, anti-aging treatments that prolong old-age will probably increase economic strain in terms medical costs and familial demand.
There are also other potential benefits to longer healthy lives:
- If people have more time alive, they have more slack to take risks and try things that they're interested in
- Academic problems that demand an unusually large time investment to fully understand may be tractable. No idea whether these actually exist or not but Scott Alexander wrote a cool story about it. Progress may get faster
- People care about their own welfare so people who live longer may care about worldly conditions over a longer time-frame than they did previously. This could make people think slightly more long-term about their decisions
But how does this compare to just having more kids?
You mention that raising a kid costs $234K. But adults, even frugal ones, may have around $25K annual expenses. This implies that keeping an adult around for the same amount of time (17 years) costs upwards of $425K. This would be more resource intensive if those adults were not working. If we account for the fact that adults pay their own expenses and kids don't, funding adults life-years strikes me as more economical than paying for more kids. The implication here is that, not only do adults demand fewer resources than kids but they also contribute more to the economy (in spending) than them.
So, overall, I'd guess that life-extension would be an economic benefit. However a lot of these factors are very culturally dependent and the ways that life-extension would affect culture could vary.