Remove @GuardedBy annotations from SettingsRegistry members

... as SettingsRegistry object is already guarded by a lock
in the outer class (SettingsProvider). This change also performs
some minor cleanups within SettingsProvider.

Test: atest SettingsProviderTest
Bug: 371801645
Flag: EXEMPT minor cleanup
Change-Id: I5946e50264323a5410f1bed3f5773a9a5c3bb429