I keep my bios.xml file unchanged, I use the correct BIOS, and everything works:

<bios path="scph1001.bin" md5="dc2b9bf8da62ec93e868cfd29f0d067d,924e392ed05558ffdb115408c263dccf" core="pcsx_rearmed" note="Recommended for better compatibility." />

Now warning message is not shown but any PSX game I load, it just kick me out (just as before)...

You are "cheating" the MD5 checker, but that is not enough to "cheat" the Emulator: If you have a problem with the BIOS, or you are using the wrong MD5, or you have the wrong name or extension (remembering that they are case sensitive, name and extension).

This is really kind of a hassle. Why include the emulator when it's broken?

It is not because you were unable to make it work that it is broken, I use this emulator without any problems, and I did not have to make any configuration or modifications to work, it was enough to add the correct BIOS.

@warriorfr @acenet @IlNiCoLiNo @UdderJuice
I recommend that you use the latest official version of Recalbox (do not use ready-made images from the internet, or outdated versions), and use the correct BIOS files (remembering that these files are copyrighted, so requesting or distributing them is against the forum rules).

This post apparently refers to an earlier version of Recalbox, this procedure does not currently make sense, do not get confused.