It's a Mitsubishi Heavy Industries split system. The two options I had were an IR blaster or a DIY ESPhome-based module plugged directly into the unit that controls it over the SPI bus. I opted for the latter as it gives full status info in addition to control.
I've also got a Samsung unit in another room that I can control. For that one I use SmartThings... not ideal as it goes through the cloud, but I'll take what I can get.
If you've got an old-school heater, you might have luck with some of the smart thermostats designed to be retrofitted into old houses.
Edit: just looked up your heaters online. Since you've got a lot of them, and they look pretty old, I'm guessing the smart controllers are just acting as relays. So yeah perhaps an ESP32 relay module would be the way to go! Once you've got the code working for one, you could roll them out to the rest. You'd need some confidence working with relays and electronics of course.