Provide LocalKey to every custom transition (#1869)

Fixes #1551.

Also, remove superfluous `debugPrint` calls and other `Logger` setup
since the latest `pkg:logging` does this stuff automatically and so the
setup lead to duplicate messages in console.

## Pre-launch Checklist

- [x] I read the [Flutter Style Guide] _recently_, and have followed its
advice.
- [x] I signed the [CLA].
- [x] I read the [Contributors Guide].
- [x] I updated/added relevant documentation (doc comments with `///`).
- [x] All existing and new tests are passing.

<!-- Links -->
[Flutter Style Guide]:
https://github.com/flutter/flutter/wiki/Style-guide-for-Flutter-repo
[CLA]: https://cla.developers.google.com/
[Discord]: https://github.com/flutter/flutter/wiki/Chat
[Contributors Guide]:
https://github.com/flutter/samples/blob/main/CONTRIBUTING.md

game_template/lib/main.dart

@@ -8,7 +8,6 @@
 // import 'firebase_options.dart';
 
 import 'package:firebase_crashlytics/firebase_crashlytics.dart';
-import 'package:flutter/foundation.dart';
 import 'package:flutter/material.dart';
 import 'package:flutter/services.dart';
 import 'package:go_router/go_router.dart';
@@ -64,16 +63,6 @@ Future<void> main() async {
 
 /// Without logging and crash reporting, this would be `void main()`.
 void guardedMain() {
-  if (kReleaseMode) {
-    // Don't log anything below warnings in production.
-    Logger.root.level = Level.WARNING;
-  }
-  Logger.root.onRecord.listen((record) {
-    debugPrint('${record.level.name}: ${record.time}: '
-        '${record.loggerName}: '
-        '${record.message}');
-  });
-
   WidgetsFlutterBinding.ensureInitialized();
 
   _log.info('Going full screen');
@@ -134,8 +123,10 @@ class MyApp extends StatelessWidget {
             GoRoute(
                 path: 'play',
                 pageBuilder: (context, state) => buildMyTransition<void>(
+                      key: ValueKey('play'),
                       child: const LevelSelectionScreen(
-                          key: Key('level selection')),
+                        key: Key('level selection'),
+                      ),
                       color: context.watch<Palette>().backgroundLevelSelection,
                     ),
                 routes: [
@@ -147,6 +138,7 @@ class MyApp extends StatelessWidget {
                       final level = gameLevels
                           .singleWhere((e) => e.number == levelNumber);
                       return buildMyTransition<void>(
+                        key: ValueKey('level'),
                         child: PlaySessionScreen(
                           level,
                           key: const Key('play session'),
@@ -172,6 +164,7 @@ class MyApp extends StatelessWidget {
                       final score = map['score'] as Score;
 
                       return buildMyTransition<void>(
+                        key: ValueKey('won'),
                         child: WinGameScreen(
                           score: score,
                           key: const Key('win game'),