HomePod mini Setup Failed

I have been trying to setup a new HomePod Mini, but it's been failing repeatedly with the message "Setup Failed (100)".

Here's what happens:

1. I power up my new HomePod mini.
2. My iPhone Pro 13 (w/ iOS 16.1, and all apps up-to-date) detects it, and I press the "Set Up" button to set it up.
3. Next I'm asked to "Center the HomePod in the Frame", which I do. The HomePod bongs accordingly.
4. I'm then asked to "Pick a Room", so I select my choice and continue.
5. I'm then asked if I want "Personal Requests". I've tried both answers, on different occasions, with the same results.
6. The iPhone then produces a "Setting Up" status.
7. After a short while, I'm asked for my Apple ID Password, which I enter. (Note I have 2FA setup, but I'm not asked for a code to setup the device. I'm assuming that this is normal.)
8. Shortly after that, I get a failure message. This message changes error code. Sometimes it's "Setup Failed (100)", sometimes it's "Setup Failed (-1001)". Once, it seemed to succeed, by my Home app stated that it was "Configuring" the HomePod, and it stayed in that state, and completely unusable, for over a day.

I've tried rebooting my iPhone, resetting the HomePod, and cursing. Nothing has worked so far.

Any suggestions?