Unfortunately the only way I’ve been able to get my HomePod Minis to work reliably is to disable the 5Ghz band completely and only have 2.4 — even when they’re in the same room as the AP!
Having two separate SSIDs and joining them to the 2.4GHz SSID also works, but it’s kind of a pain and you lose certain functionality if they aren’t on the same SSID as your iPhone.
Interesting to read you have issues with your HomePod Mini. Mine has been stable, however all my (7) original HomePods had constant issues connecting starting around OS 14.5 / 15. I have to imagine it's related to AirPlay 2.
I went through two different WiFi-6 networks, in every possible configuration, with no luck. Cumulatively I must have spent 20-30 hours debugging the networks, and finally happened upon something that works, albeit with caveats. I got two Eero Pro 6 access points, and disabled band steering — it was an experimental setting at the time — and everything worked! I've since re-enabled the setting and immediately have issues so I feel confident saying band steering is the issue.
Since then Eero has updated the setting to include AP switching and it's called "client steering". So, unfortunately, to have my HomePods connect I sacrifice AP handoff and have to toggle wifi if I move my laptop to the other end of the apartment.
There are quite a few folks who have brought this up in an Apple issue, and I have a friend in Tucson — where radio interference isn't an issue — with the same problem.
I have in my router options possibility to choose what Gz should device connect to: for example, old notebook I connect to 2.4 only. Maybe you should enable same for homepod?
The SSID should’t matter if they are on the same IP address space. Make sure your router isn’t handing out different DHCP ranges for each SSID or frequency range.
Having two separate SSIDs and joining them to the 2.4GHz SSID also works, but it’s kind of a pain and you lose certain functionality if they aren’t on the same SSID as your iPhone.