Revert "Avoid locks in various singleton HandlerThread types"

This reverts commit 198ebf4b.

Reason for revert: Breaks some Robolectric tests that have fragile
assumptions about internals and reflections. Will fix when relanding.

Bug: 411249866
Bug: 414673812
Change-Id: I8dc7161992a493c1ccf5952e7dadb8f9c5f539e5