It mostly duplicates code of do_handle_one_connection(). Move it there,
so that it is easier to simplify it in future.

Moved PSI_CALL_delete_current_thread()/_db_pop_() out of mutex. Call them
consistently in all cases, unlike before:
- called for reused threads
- called for flushed threads
- called for threads that go down after 5 minutes timeout
- not called for threads attempting to cache themselves during flush
- not called for threads attempting to cache themselves when cache is full

Part of MDEV-19515 - Improve connect speed