The new HomePod Minis did need to be updated, but that didn't solve the problem. I did all sorts of things, like restarting the HomePods, removing them from Home and then re-adding them, making sure I was correctly logged into Apple Music, turning off my phone. Eventually it started working for most playlists, except for the biggest playlist I have, which really is probably too big. I'm wondering if there was some sort of iCloud syncing thing that took time. But I'm not sure.
In general, I'm finding the HomePod Minis disappointingly glitchy. They can work fine most of the day, but every so often one of several things goes wrong: one speaker turns off and then comes back on after several seconds (or doesn't—that was also mostly solved by a router restart), the HomePod stereo pair suddenly invites my old HomePod to join in (without me doing anything to initiate that), it still won't play my longest playlist when I try to start it via voice command (hey, Siri...), but it will usually do find when I use AirPlay with my iPhone or iPad and the Apple Music app. But then sometimes, a playlist will be playing audibly, but both the Home app and the Apple Music app say that nothing is playing. There are more oddities that go on, but I can't remember them all. Usually something basic, like turning off my phone or restarting the router will help.
It's annoying, though. I'm old enough to remember when much "less advanced" technologies worked right out of the box and kept working for a long time without any need to call customer service. Feeling nostalgic...