Repository: flutter
Issue: image.network Incorrect user-agent
Labels: framework,a: images,has reproducible steps,P3,team-framework,triaged-framework,found in release: 3.13,found in release: 3.15

### Is there an existing issue for this?

- [X] I have searched the [existing issues](https://github.com/flutter/flutter/issues)
- [X] I have read the [guide to filing a bug](https://flutter.dev/docs/resources/bug-reports)

### Steps to reproduce

The user-agent of image.network is incorrect. After specifying, there will be an extra default user-agent character in front.

### Expected results

user-agent is `Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/117.0.0.0 Safari/537.36`

### Actual results

user-agent is `Dart/3.1 (dart:io), Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/117.0.0.0 Safari/537.36`

### Code sample

<details><summary>Code sample</summary>

```dart
import 'package:flutter/material.dart';

void main() {
  runApp(const MyApp());
}

class MyApp extends StatelessWidget {
  const MyApp({super.key});

  // This widget is the root of your application.
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
        title: 'Flutter Demo',
        theme: ThemeData(
          useMaterial3: true,
        ),
        home: Scaffold(
          appBar: AppBar(
            backgroundColor: Theme.of(context).colorScheme.inversePrimary,
            title: const Text("image network test"),
          ),
          body: Center(
              child: Container(
                  height: 300,
                  width: MediaQuery.of(context).size.width / 2,
                  child: Image.network(
                    "https://files.flutter-io.cn/images/branding/flutterlogo/flutter-cn-logo.png",
                    headers: const {
                      "user-agent":
                          "Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/117.0.0.0 Safari/537.36"
                    },
                  ))),
        ));
  }
}

```

</details>


### Screenshots or Video

<details>
<summary>Screenshots / Video demonstration</summary>

![2023-09-30](https://github.com/flutter/flutter/assets/70017178/8c55390d-a772-4113-bdcf-cbb9f2bc333e)


</details>


### Logs

no response


### Flutter Doctor output

<details><summary>Doctor output</summary>

```console
[!] Flutter (Channel stable, 3.13.4-0.0.pre.1, on Microsoft Windows [版本 10.0.22621.2283], locale
    zh-CN)
    • Flutter version 3.13.4-0.0.pre.1 on channel stable at C:\tools\flutter
    ! Warning: `dart` on your path resolves to C:\tools\dart-sdk\bin\dart.exe, which is not inside
      your current Flutter SDK checkout at C:\tools\flutter. Consider adding C:\tools\flutter\bin to
      the front of your path.
    • Upstream repository https://github.com/flutter/flutter.git
    • Framework revision a625ad4ea9 (3 weeks ago), 2023-09-10 02:15:21 +0200
    • Engine revision b8d35810e9
    • Dart version 3.1.1
    • DevTools version 2.25.0
    • If those were intentional, you can disregard the above warnings; however it is recommended to
      use "git" directly to perform update checks and upgrades.

[✓] Windows Version (Installed version of Windows is version 10 or higher)

[!] Android toolchain - develop for Android devices (Android SDK version 33.0.0)
    • Android SDK at F:\android\sdk
    ✗ cmdline-tools component is missing
      Run `path/to/sdkmanager --install "cmdline-tools;latest"`
      See https://developer.android.com/studio/command-line for more details.
    ✗ Android license status unknown.
      Run `flutter doctor --android-licenses` to accept the SDK licenses.
      See https://flutter.dev/docs/get-started/install/windows#android-setup for more details.

[✓] Chrome - develop for the web
    • Chrome at C:\Program Files\Google\Chrome\Application\chrome.exe

[✗] Visual Studio - develop Windows apps
    ✗ Visual Studio not installed; this is necessary to develop Windows apps.
      Download at https://visualstudio.microsoft.com/downloads/.
      Please install the "Desktop development with C++" workload, including all of its default
      components

[✓] Android Studio (version 2021.2)
    • Android Studio at C:\Program Files\Android\Android Studio
    • 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+7-b1504.28-7817840)

[✓] IntelliJ IDEA Community Edition (version 2022.2)
    • IntelliJ at C:\Users\a3873\AppData\Local\JetBrains\Toolbox\apps\IDEA-C\ch-0\222.4345.14
    • 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

[✓] VS Code (version 1.82.2)
    • VS Code at C:\Users\a3873\AppData\Local\Programs\Microsoft VS Code
    • Flutter extension can be installed from:
      🔨 https://marketplace.visualstudio.com/items?itemName=Dart-Code.flutter

[✓] Connected device (3 available)
    • Windows (desktop) • windows • windows-x64    • Microsoft Windows [版本 10.0.22621.2283]
    • Chrome (web)      • chrome  • web-javascript • Google Chrome 117.0.5938.132
    • Edge (web)        • edge    • web-javascript • Microsoft Edge 117.0.2045.43

[✓] Network resources
    • All expected network resources are available.

! Doctor found issues in 3 categories.
```

</details>