Kick off health checks with lock held.

We've seen evidence from Log.wtf() that health checks are racing with
other mutations because the AMS lock isn't held.  To mitigate this,
perform health checks through our Handler where the lock is acquired.

Bug: 267673062
Test: treehugger
Change-Id: I8c7411bc42475789373c29c57cbd384faaf7b9ee