Extract the not-strictly-modes-related parts of ZenModesBackend

So that when it's moved to SettingsLib, it doesn't need to carry that baggage.

Bug: 346519570
Test: atest com.android.settings.notification.modes
Flag: android.app.modes_ui
Change-Id: I7911a521d96f5dbac2c2395171d324b7b54b8b07