My Apple ID section in settings is grayed out

If an app or feature on your device is not working, it may have been disabled. See how to 'Use Screen Time on an iPhone, iPad, or iPod touch' --> https://support.apple.com/HT208982

Have you tried disabling screen time to see if you can log into iCloud?

This answer is not helpful. We are experiencing the same issue on my child’s device. iCloud is greyed out and screen time is not working as it should. She’s overriding the time limits.

I cannot get her iCloud turned on.

I did and it did work.

I should have clarified, the instructions above needed to be more clear that there is a conflict between iCloud and screen time sometimes.

I appreciate the help with both as my daughter’s screen time rules haven’t been working as they should anyway. Hopefully this resolves both issues.

You are welcome.