[tool] Use 'flutter pub get' for Flutter packages (#4397)

Extracts common logic for running `pub get`, and switches commands to use it. The common logic always uses `flutter pub get` for Flutter packages, rather than `dart pub get`, since the latter will fail if someone has a non-Flutter `dart` in their path before `flutter` (e.g., Dart team members contributing PRs).

script/tool/test/update_dependency_command_test.dart

@@ -429,7 +429,7 @@ dev_dependencies:
       expect(
         output,
         containsAllInOrder(<Matcher>[
-          contains('dart pub get failed'),
+          contains('Fetching dependencies failed'),
           contains('Failed to update pigeon files'),
         ]),
       );
@@ -545,7 +545,7 @@ dev_dependencies:
       expect(
         output,
         containsAllInOrder(<Matcher>[
-          contains('dart pub get failed'),
+          contains('Fetching dependencies failed'),
           contains('Failed to update mocks'),
         ]),
       );