AirPlay is the worst Apple product that I’ve ever had to use on a regular basis, and I say this as someone that still has one of *those* MacBook Pro keyboards.
Will it work at all? If it does, will it then stop when the screen locks? Neither of these seems to have a consistent answer, it’s a total crapshoot.
Actually Screen Time is total shit as well. Might even be worse. Certainly it’s utterly unreliable in syncing changes and working as expected.