Hi Jakub,
Man, getting hit with error codes like 0x80073cf6 and 0x80548225 just when you're trying to play Forza Horizon 6 is the absolute worst. As you mentioned, since the Gaming Services repair tool and standard reinstalls are looping, the Microsoft Store registration is completely choked up.
Since you’ve already tried the XboxInstaller.exe route, you usually have to nuke the Gaming Services registry keys to force Windows to pull a clean copy. Give this a shot:
Right-click the Start Menu, open PowerShell (Admin), and run this command to completely wipe the broken service: get-appxpackage *gamingoperators* | remove-appxpackage -allusers
Open the Registry Editor (cmd -> regedit) and navigate to: HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services
Delete the GamingServices and GamingServicesNet folders if they are still there.
Restart your PC, open the Xbox App, and it should finally let you install a working version of Gaming Services without throwing those error codes.
Let me know if that finally gets you past the loading screen!