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

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

The original change was reverted due to flakiness it introduced in tests that use fling gestures.
* https://github.com/flutter/flutter/issues/135728

It was reverted again due to a change in the leak tracking tests that broke it.