[QuickStart] Add GetFeatureSupportStatusAsync

* Report the feature support status via callback.
* Implement the callbacks logic in the super class to cover both fake
and real implementation
* Add initial_support_status into
FakeTargetDeviceConnectionBroker::Factory for testing
* Tweak the browsertest to cover the async logic


Bug: b:238849500
Change-Id: I85417cdb70d849ef8abe336a54217ea13881f724
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/3876065


Commit-Queue: Roman Sorokin <rsorokin@google.com>
Reviewed-by: Brittany Hartmire <bhartmire@google.com>
Cr-Commit-Position: refs/heads/main@{#1045763}