I got an iPhone 12 Pro for testing. In order to have all the functionality and settings cloned from my old iPhone SE (2nd gen), I used the Quick Start as described by Apple.
The new phone worked fine but in the end I decided to keep my iPhone SE and delete the iPhone 12. I followed "What to do before you sell, give away, or trade in your iPhone or iPad".
Now, all my AirTags are chiming when moved once a day. The AirTags' locations still display fine (i.e. "With you") in the FindMy App, both on my Mac and iPhone but when clicking "play sound", it says "Item Not Reachable" in spite of being next to them with my phone with bluetooth turned on.
So the AirTags are apparently still linked to my iCloud account but not to my iPhone SE anymore.
Did I miss any advice from Apple? How can I re-link the AirTags to my iPhone SE with minimal efforts so they don't chime once a day when moved and can be pinged by my iPhone?