Hello asdasdasd,
The two Gaming Services folders do not necessarily indicate a conflict. They appear to be different components of the same Gaming Services package rather than duplicate installations, so I would not recommend manually deleting either folder from the protected WindowsApps directory.
While 10 GB should normally be enough for the Gaming Services package itself, freeing some additional space may still help because Microsoft Store installations can require temporary space during downloading and installation.
If you use a third-party antivirus program, temporarily disable its real-time protection and retry the installation once. Re-enable the protection immediately afterward, regardless of the result. If you only use Microsoft Defender, leave it enabled.
As an additional test, create a new local Windows user account, sign in to it, and try installing Gaming Services through the Microsoft Store. If it works in the new account, the issue may be limited to the Store or app registration in your current Windows profile.
If the issue continues in the new Windows profile, open Command Prompt as administrator and run these commands one at a time:
DISM.exe /Online /Cleanup-Image /RestoreHealth
sfc /scannow
After both scans finish, restart the computer and try installing Gaming Services again.
Please let me know how it goes after trying these steps, and I'll be glad to continue looking into it with you.