This pull request is created by [automatic cherry pick workflow](https://github.com/flutter/flutter/wiki/Flutter-Cherrypick-Process#automatically-creates-a-cherry-pick-request)
Please fill in the form below, and a flutter domain expert will evaluate this cherry pick request.

### Issue Link:
What is the link to the issue this cherry-pick is addressing?

DevTools release 2.36.0: https://dart-review.googlesource.com/c/sdk/+/368762

### Changelog Description:

This change adds a service extension that DevTools uses to support a "Track widget build counts" feature. This feature is first included in DevTools 2.36.0.

### Impact Description:

This service extension is required for DevTools 2.36.0, which needs to be [cherry-picked](https://dart-review.googlesource.com/c/sdk/+/368762) into the Dart SDK. This beta release of DevTools missed the code cutoff because we were targeting the date listed on go/dash-team-releases (5/29). Since the beta release went out early, we missed the cut off and need to cherry-pick.

Since this DevTools release depended on the changes from https://github.com/flutter/flutter/pull/148702, we also need to cherry pick those changes into the flutter beta channel.

### Workaround:
No. This is required for DevTools 2.36.0 to work properly: https://dart-review.googlesource.com/c/sdk/+/368762.

### Risk:
What is the risk level of this cherry-pick?

### Test Coverage:
Are you confident that your fix is well-tested by automated tests?

### Validation Steps:
What are the steps to validate that this fix works?

With this change, a flutter app connecting to DevTools 2.36.0 (already on the Dart SDK main branch and trying to CP into the Dart SDK beta branch) will be able to use the "Track widget build counts" feature in DevTools.