See my above post.
Components -> Windows Error Reporting Service -> Problem Reports and Solutions Control Panel Support must be kept for Windows 7 Backup and Restore to remain working. Now that I believe I've found the root cause, I will go ahead with the GamerOS preset and continue testing.
Maybe this is worthy of a bug report or something?
I tested these changes with the GamerOS 10 Final and original GamerOS 11 presets, but still no luck. Aside from Windows Error Reporting and Volume Shadow Copy, there's something else that's being removed which is causing the failure with Windows Backup. However, the behaviour of when the failure happens is different. While the backup is attempting to interact with the Volume Shadow Copy, it seems to stall out instead of fail immediately. I will try out PhsMu's attached preset and see what's what. My guess at this point is that another unrelated component is causing the failure.
Some combination of components in Networking, Remoting & Privacy, and System are required for Windows Backup. I've tried fulling enabling each category on their own to test, and it's only when all three are checked off that Windows Backup will function. I don't know if I have the patience to figure this out... I've already reinstalled Win11 upwards of 20 times.
And just an FYI: I did find an alternative solution in Cobian Reflector as a free, reliable, feature-rich, and easy incremental backup software.
There is some unholy combination of components for Windows Backup to function. It also seems to involve Internet Connection Sharing in addition to my previous findings. I can't do anymore. I was able to still remove and disable the biggest annoyances from Windows, which I'm still happy with. Thank you Txmmy for the base preset and garlin and PhsMu for the help.