This updates the implementation to use the stopwatch from the Clock object and pipes it through to the TestWidgetsFlutterBinding so it will be kept in sync with FakeAsync.

Relands https://github.com/flutter/flutter/pull/138843 attempted to reland https://github.com/flutter/flutter/pull/137381 which attempted to reland #132291
Fixes https://github.com/flutter/flutter/issues/97761

1. The original change was reverted due to flakiness it introduced in tests that use fling gestures.
  * Using a mocked clock through the test binding fixes this now
2. It was reverted a second time because a change at tip of tree broke it, exposing memory leaks, but it was not rebased before landing. 
  * These leaks are now fixed
3. It was reverted a third time, because we were so excellently quick to revert those other times, that we did not notice the broken benchmark that only runs in postsubmit.
  * The benchmark is now fixed