First of all, don't confuse "GameInput" and its service(s) with "Gaming Services"...
Now try this (found on Reddit): https://www.reddit.com/r/WindowsHelp/comments/18bb6e0/comment/kcw2mem/?utm_source=share&utm_medium=web2x&context=3
In short, just open your installed app panel and isolate "gameinput": if you find something dated around feb-march 2023 UNINSTALL IT (it seems an old MS leftover for Win10), and cross your fingers 🤞🏻:
EDIT:
Tested this myself and on next reboot Win11 23H2 silently reinstalled it (same version 10.1.22621.3036), BUT now I have a single GameInput Service (Auto, triggered. "Restart service" when it fails) and no uninstall option in the App list: