Repository: flutter
Issue: [go_router] Creating an observer does not listen to .go() route changes
Labels: package,has reproducible steps,P2,p: go_router,team-go_router,triaged-go_router,found in release: 3.16,found in release: 3.19

### What package does this bug report belong to?

go_router

### What target platforms are you seeing this bug on?

iOS

### Have you already upgraded your packages?

Yes

### Dependency versions

<details><summary>pubspec.lock</summary>

```lock
[Paste file content here]
```

</details>


### Steps to reproduce

Hello!

I wanted to create an route path observer using extending the `NavigatorObserver` and I found that when I wanted to go to a specific location using `context.go()` does not call any of the overridable methods (like didPop, didPush, didReplace, etc.).

### Expected results

To have a way to also use `context.go()` and use a observer.

### Actual results

Nothing gets cached through the observer.

### Code sample

<details open><summary>Code sample</summary>

```dart
GoRouter(
  initialLocation: '/',
  navigatorKey: _rootNavigatorKey,
  observers: [
    GoRouterObserver(), 👈 Created observer
  ],
  routes: [
    GoRoute(
      ...
    )
    GoRoute(
      ...
    ),
  ],
);

class GoRouterObserver extends NavigatorObserver {
  @override
  void didPush(Route<dynamic> route, Route<dynamic>? previousRoute) {
    print('MyTest didPush: $route');
  }

  @override
  void didPop(Route<dynamic> route, Route<dynamic>? previousRoute) {
    print('MyTest didPop: $route');
  }

  @override
  void didRemove(Route<dynamic> route, Route<dynamic>? previousRoute) {
    print('MyTest didRemove: $route');
  }

  @override
  void didReplace({Route<dynamic>? newRoute, Route<dynamic>? oldRoute}) {
    print('MyTest didReplace: $newRoute');
  }
}
```

</details>


### Screenshots or Videos

<details open>
<summary>Screenshots / Video demonstration</summary>

[Upload media here]

</details>


### Logs

<details open><summary>Logs</summary>

```console
[Paste your logs here]
```

</details>


### Flutter Doctor output

<details open><summary>Doctor output</summary>

```console
[✓] Flutter (Channel stable, 3.16.9, on macOS 14.1.1 23B81 darwin-arm64, locale en-ES)
    • Flutter version 3.16.9 on channel stable at /Users/user/dev/flutter
    • Upstream repository https://github.com/flutter/flutter.git
    • Framework revision 41456452f2 (7 days ago), 2024-01-25 10:06:23 -0800
    • Engine revision f40e976bed
    • Dart version 3.2.6
    • DevTools version 2.28.5

[✓] Android toolchain - develop for Android devices (Android SDK version 34.0.0)
    • Android SDK at /Users/user/Library/Android/sdk
    • Platform android-34, build-tools 34.0.0
    • Java binary at: /Applications/Android Studio.app/Contents/jbr/Contents/Home/bin/java
    • Java version OpenJDK Runtime Environment (build 17.0.6+0-17.0.6b829.9-10027231)
    • All Android licenses accepted.

[✓] Xcode - develop for iOS and macOS (Xcode 15.2)
    • Xcode at /Applications/Xcode.app/Contents/Developer
    • Build 15C500b
    • CocoaPods version 1.14.3

[✓] Chrome - develop for the web
    • Chrome at /Applications/Google Chrome.app/Contents/MacOS/Google Chrome

[✓] Android Studio (version 2022.3)
    • 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 17.0.6+0-17.0.6b829.9-10027231)

[✓] VS Code (version 1.85.2)
    • VS Code at /Applications/Visual Studio Code.app/Contents
    • Flutter extension version 3.80.0

[✓] Connected device (2 available)
    • macOS (desktop) • macos  • darwin-arm64   • macOS 14.1.1 23B81 darwin-arm64
    • Chrome (web)    • chrome • web-javascript • Google Chrome 121.0.6167.85

[✓] Network resources
    • All expected network resources are available.

• No issues found!
```

</details>