Okay let's do some normal chess things today.
I replaced the finials on these kings and queens—they had been given poorly color-matched plastic beads by a previous owner and the queens had only rough and glue-stained empty sockets. Now I'm carving out the bottoms to put in weights. It looks like they used to have weights or were meant to get some eventually.
Trying to go for minimal aesthetic changes. I don't know what the finials were ever supposed to look like, but I am pleased with this solution.