Thank you for the detailed process, I will use it to benefit of the sideloaded mode.
Well, I finally was able to repair the thing. I decided to try one last time that manual factory reset thing and the basic sqlite statement given by Rashkae, and... it worked !
So it is now sure the issue was a broken database with unnecessary modifications following complicated sqlite statements found in various places. There is actually no wiki page about registration bypassing, but a link sending to that topic
https://www.mobileread.com/forums/sh...d.php?t=171664. I think a page should be created with the simple command indicated by Rashkae and a statement telling people to be careful with other solutions. I will not do it now because the topic of modifying the database seems quite hot, and doing so could make some people angry.
What it actually does is creating another row in the user table, with default values for every entry. The default row generated during the factory reset process is kept unmodified.
I wish to thank everybody who took time to answer me, as this led to the resolution of my issue and I learned new things about my device. I will now flash it and go on learning things about it.