I passed the same problem with Mathematica 12.3. I use Office with Microsoft 365, and the next monthly update of Office reinstalled Math Recognizer. Mathematica 12.3 and 13 kept running fine. I use Windows 11.

IIRC, you need to uninstall the Math Recognizer from Settings app > Apps > Optional features. Reboot and the application should start (maybe slowly but successfully).

Ok, but does Mathematica 12.1 at least install without any issues?

I believe 12.1 behaves in a similar manner to 12.2. My memory is a bit fuzzy here, it has been a while.

When I tried to launch Mathematica in Windows 10 WSL2 Ubuntu on Surface Pro X (without Insider´s Preview), it installed correctly from Shell (sh) script, but upon running there was an error, that I cannot run AMD64 architecture file on aarch64 architecture, although the build was for Linux.

The x64 emulator is for Windows executables. i.e., it does not apply to x64 binaries running in WSL.

I haven't tried 12.3, but 12.2 would not start for me in the x64 emulator. If you are interested in Windows ARM64 support, you should email Tech Support to let them know of your interest in it.