Thank you, this is news to me, and apparently something is stubbornly not working properly in my setup. That "indefinite" option, which I've never seen before but you kindly highlight, should fix this case. Time to... debug a coin.
If you are seeing their name (ie "John's AirTag" or "John's iPhone") then it should work. On the AirPods I mentioned that don't work, no name appears which I suspect is why.