Yeah, that's what I thought, but I was hoping for a solution. I know a couple Sonos products have a line in, which they can then beam to other Sonos speakers. And Apple's HomePods support multi-room audio over airplay, so in theory you could use a Raspberry Pi with an open source airplay server to feed to that (although I don't know if you need Airplay 2 for multi-room audio). Of course both Sonos and HomePod are MUCH more expensive than what I was planning, so definitely a no-go on those.
Oh well, I might go the bluetooth transmitter route so I can at least stream wirelessly to one speaker.