On the other hand, I have done some more research and determined that I can probably fix the existing chair for about $25 in materials and a lot of sweat. Or I can hire someone to do it for some indeterminate larger amount and get it done in finite time.
(It turns out that the replacement part I need is called a "chair spreader", which I found out when I did some more poking around on the Internet. The piece I need is *remarkably* standard, along with a Forstner bit of the correct size to drill out a mating hole since there's an extra crossbar support on this design of chair. And varnish. Varnish is important...)
Gretchen is looking for a chair repairman. We've found one fellow; we'll see what we hear back.