So this is about two weeks later. iOS16.4 came out and I manually updated all affected iDevices to 16.4 on March 28. Unbeknown to me, the update reset our choices of "Automatic updates = off" to "on", both on normal settings and the Home app settings. As you guessed, both the regular HomePod and the latest Homepod Mini got the 16.4 updates (rats!)
The regular HomePod continues to be unaffected, but the HomePod Mini reverted to the "...one moment (or "wait a sec"), still working, I'm having trouble connecting to the Internet..." behavior.
Since I've had no success with Apple Support other than using my time and theirs and getting replacement HomePod Minis under AppleCare is getting old. So...
- ...I went to the Home app and looked under Home Settings, Wi-Fi Networks & Routers, and turned off HomeKit Accessory Security.
- Immediately after this, I gave Siri commands via the HomePod Mini, and every one of the 10 or so was successfully completed.
Later on the same day, the HomePod Mini went back to the same "no can do" behavior. While both Apple Support and I had looked at my eero mesh system settings and found nothing inherently incorrect...
- ...after seeing a post in eero support about mesh systems and HomeKit, I looked into our eero's "Discover settings", eero Labs (BETA) and turned off the WPA3 security settings
- A you guessed, immediately after this, I gave Siri commands via the HomePod Mini, and every one of the 6 or so was successfully completed.
On this second day, the behavior of the HomePod Mini to Siri requests is similar to some of the responses in this thread, mostly successful, some not. While this is not what should be happening, I am worn out on this issue. This may be as good as it gets. A sad performance for an otherwise good Apple product.