Update ! and bad news !
I did a complete reset of my MacBook Air, erase the DD, dowloaded a fresh Big Sur and created two users (without using Time Machine).
So I have MacBook like new, not a single extra piece of software or driver or data, two empty users we will call A & B. My plan was to slowly add software and data and drivers until the moment I will see the bug appear, it would have given me an indication of what the problem could be. Things got faster than I expected ! 😤
I then setup an email account for user A, not an iCloud account but a simple map account. I give it time to download emails and use the search in Mail, it works as expected.
I go to user B, I don't even create an email account, I use fast switching to User A and Mail search still works as expected.
I fast switch back to user B, do nothing and simply log out of user B, remember this is what triggers the bug.
Log in user A, and bam, search is broken, exactly like it happens on my two Macs since Big Sur.
On an empty MacBook with only Big Sur installed, no other piece of software !
It means that he bug does not depend on anything else than Big Sur and Mail, not even iCloud, 100% reproducible and affecting every Mac with multiple users ? I cannot believe this went through QA in the first place and 3 months after the release of Big Sur it still exists ?
Apple, please, don't make Big Sur the Vista of Apple, I hate to realise I spent hours trying to understand why this bug happens thinking it is because of my configuration, only to find out it is a huge atrocious bug in Big Sur from the start.
So disappointed, and no answers to the multiple feedbacks I reported.
So sad...