This reverts commit f5ac225c, i.e. https://github.com/flutter/flutter/pull/137115.

This is a continuation of https://github.com/flutter/flutter/pull/142266 that was redone based on feedback to make this easier to revert in the future. The exact steps I took to create this revert:

1. Revert commit noted above
2. Fix merge conflicts, that notably involved reverting some changes in https://github.com/flutter/flutter/pull/140744 ~and https://github.com/flutter/flutter/pull/141417~ (fixed my merge to avoid the second PR from being affected)
3. Delete `packages/flutter_tools/test/integration.shard/android_plugin_skip_unsupported_test.dart` as this was added in the commit noted above

cc @Gustl22 since I couldn't tag as a reviewer