Extract `ServiceListing` code to `AppsNotificationAccessScreen`

An unrelated part of `ServiceListing` was creating an
`android.os.Handler`, which was breaking the `PreferenceServiceTest`
tests since the setup code isn't run on the main Android thread.

Bug: 415243959, 415926645
Test: adb shell cmd app_function execute-app-function --package com.android.settings --function getPermissionsDeviceState --parameters {} | grep notification_access -C 20
Flag: com.android.settings.flags.device_state
Change-Id: I4c5b6e6234baff5c3f7fad2c5454d327ffbf86e9