Reland migration (#69996)

This reverts ae0a9cb560ff3b334ab0cc23980374792888b2d1 and therefore relands #69629.

Additionally, `flutter update-packages --force-upgrade` has been run to update `pubspec.yaml` to resolve the analyze test failures.
diff --git a/dev/bots/test.dart b/dev/bots/test.dart
index 5907e59..89f59d0 100644
--- a/dev/bots/test.dart
+++ b/dev/bots/test.dart
@@ -674,6 +674,7 @@
     await _pubRunTest(path.join(flutterRoot, 'dev', 'devicelab'), tableData: bigqueryApi?.tabledata);
     await _pubRunTest(path.join(flutterRoot, 'dev', 'snippets'), tableData: bigqueryApi?.tabledata);
     await _pubRunTest(path.join(flutterRoot, 'dev', 'tools'), tableData: bigqueryApi?.tabledata);
+    await _pubRunTest(path.join(flutterRoot, 'dev', 'benchmarks', 'metrics_center'), tableData: bigqueryApi?.tabledata);
     await _runFlutterTest(path.join(flutterRoot, 'dev', 'integration_tests', 'android_semantics_testing'), tableData: bigqueryApi?.tabledata);
     await _runFlutterTest(path.join(flutterRoot, 'dev', 'manual_tests'), tableData: bigqueryApi?.tabledata);
     await _runFlutterTest(path.join(flutterRoot, 'dev', 'tools', 'vitool'), tableData: bigqueryApi?.tabledata);