Touch ID on Magic Keyboard stopped working after updating to Ventura

It was working perfectly fine on Monterey and many months ago, after updating to Ventura, it just stopped working properly, unless... it is always wired to the Mac via the lightning cable. I tried everything I found online: deleting it from the bluetooth setting and reconnecting it solves it only temporary, until the Mac goes to sleep. This is a good example of how not always Apple products "just work"!

If there is any other solution to make it work again, I'm glat to try. Thank you!