In Beta 4 of MacOS 15, the iPhone Mirroring settings now has a "Reset iPhone Access" button. This fixes the problem.

Bump. I have looked all over to see how to reset this. At first, I thought that maybe there were credentials hidden in keychain, but I removed anything that looked like it might be related. On the iPhone, there is nothing in settings that seems to be related to the feature, and this isn't an app on the iPhone you can just open. When you first set this up, it asks you to login on the iPhone but I can't find a way to force that to happen again. There must be some kind of token that gets cached on the phone.

So, I lucked out. I tried again, but had to walk away for a few hours, so I just left it on a charger. When I got back, I was finally back. I am thinking that there may be a resource on the backend that got overwhelmed, and after enough pinging, it finally managed to get the right signal from Apple.

Me too. I went back and did a recovery (presumably to 17.2.x) and in order to restore my backup it updated to 17.3 Beta 2 and it all happened again.