Alert engine upon registering ServiceBinding (flutter#126075)

Send a platform message to the engine when the `ServiceBinding` is
registered. Framework side of
flutter/engine#41733

Addresses flutter#126033

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- [x] I listed at least one issue that this PR fixes in the description
above.
- [x] I updated/added relevant documentation (doc comments with `///`).
- [ ] I added new tests to check the change I am making, or this PR is
[test-exempt].
- [ ] All existing and new tests are passing.

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---------

Co-authored-by: Greg Spencer <gspencergoog@users.noreply.github.com>

packages/flutter/lib/src/services/binding.dart

@@ -45,6 +45,7 @@ mixin ServicesBinding on BindingBase, SchedulerBinding {
     SystemChannels.platform.setMethodCallHandler(_handlePlatformMessage);
     TextInput.ensureInitialized();
     readInitialLifecycleStateFromNativeWindow();
+    initializationComplete();
   }
 
   /// The current [ServicesBinding], if one has been created.
@@ -415,6 +416,14 @@ mixin ServicesBinding on BindingBase, SchedulerBinding {
   void setSystemUiChangeCallback(SystemUiChangeCallback? callback) {
     _systemUiChangeCallback = callback;
   }
+
+  /// Alert the engine that the binding is registered. This instructs the engine to
+  /// register its top level window handler on Windows. This signals that the app
+  /// is able to process "System.requestAppExit" signals from the engine.
+  @protected
+  Future<void> initializationComplete() async {
+    await SystemChannels.platform.invokeMethod('System.initializationComplete');
+  }
 }
 
 /// Signature for listening to changes in the [SystemUiMode].

packages/flutter/lib/src/services/system_channels.dart

@@ -133,6 +133,10 @@ abstract final class SystemChannels {
   ///  * `System.requestAppExit`: The application has requested that it be
   ///    terminated. See [ServicesBinding.exitApplication].
   ///
+  ///  * `System.initializationComplete`: Indicate to the engine the
+  ///    initialization of a binding that may, among other tasks, register a
+  ///    handler for application exit attempts.
+  ///
   /// Calls to methods that are not implemented on the shell side are ignored
   /// (so it is safe to call methods when the relevant plugin might be missing).
   static const MethodChannel platform = OptionalMethodChannel(

packages/flutter/test/services/binding_lifecycle_test.dart

@@ -0,0 +1,45 @@
+// Copyright 2014 The Flutter Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style license that can be
+// found in the LICENSE file.
+
+import 'package:flutter/foundation.dart';
+import 'package:flutter/scheduler.dart';
+import 'package:flutter/services.dart';
+import 'package:flutter_test/flutter_test.dart';
+
+class _TestBinding extends BindingBase with SchedulerBinding, ServicesBinding {
+  @override
+  Future<void> initializationComplete() async {
+    return super.initializationComplete();
+  }
+
+  @override
+  TestDefaultBinaryMessenger get defaultBinaryMessenger => super.defaultBinaryMessenger as TestDefaultBinaryMessenger;
+
+  @override
+  TestDefaultBinaryMessenger createBinaryMessenger() {
+    Future<ByteData?> keyboardHandler(ByteData? message) async {
+      return const StandardMethodCodec().encodeSuccessEnvelope(<int, int>{1:1});
+    }
+    return TestDefaultBinaryMessenger(
+      super.createBinaryMessenger(),
+      outboundHandlers: <String, MessageHandler>{'flutter/keyboard': keyboardHandler},
+    );
+  }
+}
+
+void main() {
+  final _TestBinding binding = _TestBinding();
+
+  test('can send message on completion of binding initialization', () async {
+    bool called = false;
+    binding.defaultBinaryMessenger.setMockMethodCallHandler(SystemChannels.platform, (MethodCall method) async {
+      if (method.method == 'System.initializationComplete') {
+        called = true;
+      }
+      return null;
+    });
+    await binding.initializationComplete();
+    expect(called, isTrue);
+  });
+}