iMessage asks to log in on each launch on macOS Sonoma, and login doesn't work

It's ridiculous: after updating to Sonoma, each time I launch iMessage on my Mac, it asks me to log in! I AM logged in with iCloud identifier already:

But, what's worse: I CAN'T LOGIN! First, there's a spinner that takes 3-4 minutes(!):

And then .... ta-da — a super-helpful message 'unknown error occurred'.

I'm living with it for 2 months already, since making this bad decision to upgrade to 'new video backgrounds on login screen', aka macOS Sonoma. Two or three times iMessage launched. Most of the time — it doesn't.

I'm just astonished by such low-quality errors and poor experience:

1. Inability to get credentials of already logged in user in their own OS, or at least store them from session to session.
2. Login time 4 minutes.
3. Inability to even respond in a better way than 'unknown error' written in red.

It's a totally off grade for a paid product to release such experiences. And I'm paying for that. :/