AirTag won't play sound or work with directional finding after battery replacement

I'm BACK……

At least partially answering my own question:

1. The problem did not rest with one iPhone iOS version or hardware model vs. another. I did no better trying to reset my recalcitrant AirTags and pairing them to my wife's phone than I had to my own.
2. In the end, it looks as though the problem relates somehow to the tag resetting process. Continuing to flout the axiom that continuing the same solution and expecting a different result is the fundamental definition of stupidity, I tried a few more times with the tags that seemed to be in Never-Never-land after I removed them from my AppleID (couldn't be identified as belonging to any one else, couldn't be paired to either of the phones that belonged to me, couldn't be disabled as potentially malevolent trackers), I wondered whether perhaps I wasn't being thorough enough with the battery removal/replacement "5 times is the charm" routine.

Turns out that actually WAS the problem, but I don't have a foolproof solution. I noticed this morning that when I picked up one of the misbehaving tags, it chirped at me. So, I decided to try to reset it once more. This time, instead of pressing the battery against the tag's internal contacts 5 times total, I removed the battery completely each time and waited a few seconds before re-inserting it and didn't stop doing so until, on the SEVENTH iteration of that, the chirp the tag emitted was QUALITATIVELY different from its six predecessors. When I replaced the cover, an image of an AirTag with no label immediately appeared on my nearby iPhone's screen. I went through the setup ritual with no difficulties (giving the tag a more clearly identifiable name in the process) and it immediately behaved in response to requests issued from my phone's FindMy app.

Things weren't quite as simple with the second misbehaving tag. I thought perhaps the time allotted to individual separations of the tag from its power source might be important; e.g., perhaps a capacitor in the tag needed to lose its electronic memory, so I waited an exaggerated amount of time with each chirp and release pair, but that didn't seem to help, until, magically, it did. I paid more attention to making certain the battery insertions enabled contact with each of the 3 pins on the internal ring of the tab, and although I never heard the absolutely distinguishably different "OK,now I'm a tabula rasa" chirp from that second tag, when I replaced its cover it, too, permitted pairing the tag to a piece of luggage.

So, all my tags are functional again. I still don't know for certain how to make certain that a reset procedure proceeds successfully, but it seems the magic doesn't require purchasing new AirTags.

I remain uncertain just what I did differently in my resetting attempts to enable that, but the message seems clear; i.e., "if at first you don't succeed, try, try again."