This sounds like a possible macOS 27.0 bug involving Screen Time Downtime, especially since turning Downtime off restores normal functionality.
Since restarting your Mac has not helped and Safe Mode only fixes Notification Centre, Iwould try the following steps before considering more drastic measures.
Check your Screen Time settings
Go to System Settings >> Screen Time >> Downtime and temporarily turn it off. Then check Always Allowed to make sure essential apps are permitted during Downtime.
Apple's guide: Change Always Allowed settings on Mac
Keep in mind that Always Allowed mainly controls app access, so it may not fix issues with Dock menus, permission dialogs, or accessories.
Review other Screen Time restrictions
Check your app limits and any Content & Privacy Restrictions. Change one setting at a time and test whether the problem returns when Downtime is enabled.
Apple's guide: Change Apps & Websites Restrictions settings
If you use Family Sharing or Share Across Devices, also check whether Screen Time is managed or synchronized from another device.
Check for macOS updates
Go to System Settings >> General >> Software Update and install any available compatible updates. If this is an operating system bug, a subsequent update may resolve it.
Apple's instructions: How to update macOS
Test with another user account
If the issue persists, create a temporary standard user account under System Settings >> Users & Groups. Test the affected features there, first without Downtime and then, if appropriate, with it enabled.
If the problem occurs only in your usual account, a user-specific setting may be involved. If it happens in both accounts, a broader macOS issue becomes more likely.
Keep the Safe Mode results in mind
The fact that Safe Mode restores Notification Centre but not the other features is useful diagnostic information, although it does not identify the underlying cause by itself.
If you need to repeat the test, follow Apple's instructions: How to use Safe Mode on your Mac.
Report the issue to Apple
Since disabling Downtime appears to restore normal functionality, I would report this as a suspected Screen Time bug. Include your MacBook Air M5 model, macOS 27.0 version, affected features, and the results of your restart and Safe Mode tests.
Using the Product Feedback - and make it known to Apple regarding this ongoing issue
Contact Apple Support
A short screen recording showing the affected features failing with Downtime enabled and working again after it is disabled would be especially helpful.