Repository: flutter
Issue: [Autofill] TextInput.finishAutofillContext() doesn't bring up the system prompt
Labels: a: text input,platform-ios,framework,has reproducible steps,P2,found in release: 3.3,found in release: 3.7,team-ios,triaged-ios

I do have the autofill service setup in my flutter app on iOS (including associated domains). 
The api doesn't seem to work as it's explained in the docs.  See below:

## Steps to Reproduce

1. Fill email input field
2. Fill password input field
3. Press the "Next" button which calls `TextInput.finishAutofillContext()`

**Expected results:** 

iOS prompt to save the password should appear

**Actual results:** 

iOS prompt to save the password appears after the password input loses the focus (i.e pressing the done button on the keyboard).
Pressing the "Next" button does not do anything.

<details>
<summary>Code sample</summary>

```dart
import 'package:flutter/material.dart';
import 'package:flutter/services.dart';

void main() => runApp(const MyApp());

class MyApp extends StatelessWidget {
  const MyApp({super.key});

  static const String _title = 'Flutter Code Sample';

  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: _title,
      home: Scaffold(
        appBar: AppBar(
          title: const Text(_title),
        ),
        body: const MyStatefulWidget(),
      ),
    );
  }
}

class MyStatefulWidget extends StatefulWidget {
  const MyStatefulWidget({super.key});

  @override
  State<MyStatefulWidget> createState() => _MyStatefulWidgetState();
}

class _MyStatefulWidgetState extends State<MyStatefulWidget> {
  final TextEditingController email = TextEditingController();
  final TextEditingController password = TextEditingController();

  @override
  Widget build(BuildContext context) {
    return ListView(
      children: <Widget>[
        AutofillGroup(
          onDisposeAction: AutofillContextAction.cancel,
          child: Column(
            children: <Widget>[
              TextField(
                controller: email,
                autofillHints: const <String>[AutofillHints.email],
                keyboardType: TextInputType.emailAddress,
              ),
              TextField(
                controller: password,
                autofillHints: const <String>[AutofillHints.password],
                keyboardType: TextInputType.visiblePassword,
              ),
              ElevatedButton(
                onPressed: () {
                  TextInput.finishAutofillContext(shouldSave: true);
                },
                child: const Text('Next'),
              ),
              ElevatedButton(
                onPressed: () {
                  FocusScope.of(context).unfocus();
                },
                child: const Text('Unfocus'),
              ),
            ],
          ),
        ),
      ],
    );
  }
}

```

</details>

<details>
  <summary>flutter doctor -v</summary>

```
rni-mobile % flutter doctor -v
[✓] Flutter (Channel stable, 3.3.9, on macOS 12.6 21G115 darwin-arm, locale en-US)
    • Flutter version 3.3.9 on channel stable at /Users/lukaszdebowski/flutter
    • Upstream repository https://github.com/flutter/flutter.git
    • Framework revision b8f7f1f986 (3 weeks ago), 2022-11-23 06:43:51 +0900
    • Engine revision 8f2221fbef
    • Dart version 2.18.5
    • DevTools version 2.15.0

[✓] Android toolchain - develop for Android devices (Android SDK version 33.0.0)
    • Android SDK at /Users/lukaszdebowski/Library/Android/sdk
    • Platform android-33, build-tools 33.0.0
    • Java binary at: /Applications/Android Studio.app/Contents/jre/Contents/Home/bin/java
    • Java version OpenJDK Runtime Environment (build 11.0.12+0-b1504.28-7817840)
    • All Android licenses accepted.

[✓] Xcode - develop for iOS and macOS (Xcode 14.1)
    • Xcode at /Applications/Xcode.app/Contents/Developer
    • Build 14B47b
    • CocoaPods version 1.11.3

[✓] Chrome - develop for the web
    • Chrome at /Applications/Google Chrome.app/Contents/MacOS/Google Chrome

[✓] Android Studio (version 2021.2)
    • Android Studio at /Applications/Android Studio.app/Contents
    • Flutter plugin can be installed from:
      🔨 https://plugins.jetbrains.com/plugin/9212-flutter
    • Dart plugin can be installed from:
      🔨 https://plugins.jetbrains.com/plugin/6351-dart
    • Java version OpenJDK Runtime Environment (build 11.0.12+0-b1504.28-7817840)

[✓] VS Code (version 1.74.0)
    • VS Code at /Applications/Visual Studio Code.app/Contents
    • Flutter extension version 3.54.0
```


</details>





See attached video (it shows another bug, which is the fact that the prompt did not show up at the first time at all)

https://user-images.githubusercontent.com/39137870/207063984-94eaca81-5836-47f5-8cd5-963c06fe6993.mp4

App running on iOS 16.1.2