Relanding based on this comment:
- https://github.com/flutter/flutter/pull/139278#issuecomment-1832951108

Related to tracker issue:
- https://github.com/flutter/flutter/issues/128251

<img width="278" alt="image" src="https://github.com/flutter/flutter/assets/42216813/cee7b9be-48d6-48e5-8c39-de28d0a1f0de">

The image above shows all of the instances where we have `sendTiming`. All of the call sites have been updated to use the new `Event.timing` event from `package:unified_analytics`.