Hi, @basj and @Neil Banfield
Version is 20H2 - OS build 19042.867, Cortana was disabled via WDP (+ a registry change to prevent explorer from using Bing) prior installing Start10 . And here is one big realization that just hit me; all this time I've been using a "low-key Cortana" meaning that either Start10 was installed or not, what Windows visually recalls for me to search with was, visually, a Cortana-style window.
So, after Start10 was installed, and its own searchbar setting was left un-ticked, I got the buggy research issue. I didn't realize it was there, and just kept using it the same way as usual. BUT if I enable the setting that replaces the Start10's searchbar, I get the mini-version of Cortana or rather explorer, just integrated in the Start menu that also works flawlessly! SO, this option works in the opposite way for me but actually fixes the issue.
My apologies in advance if the description (and me not behing able to change language to ENG) is a bit confusing. Guess 2 images for the sake of documentating it will explain way better than me this weird issue:


Thanks everyone for the help!