Partial Screen Sharing - Use new app chooser base class

The old one in the framework is deprecated, and not developed anymore.
Partial Screen Sharing is the only thing still using it, and preventing
it from being removed.

There also bug fixes and improvements in the new one, that the Partial
Screen Sharing feature would benefit from.

Bug: 265905535
Test: MediaProjectionAppSelectorScreenshotTest
Test: Manually build and run to trigger the app selector
Change-Id: Icbc5b3bcf641288e12fc99973d8dae3467256a2c