[DevTools 2.48.0] Network Page stops capturing requests after Clear + Restart (works on 2.42.3)

On Flutter 3.35.4 (Dart 3.9.2, DevTools 2.48.0) the Network tab in Dart DevTools stops capturing any requests after using "Clear" and then doing "Restart" in debug mode.

On the same project with Flutter 3.29.3 (Dart 3.7.2, DevTools 2.42.3) everything works correctly.
So the regression appeared between DevTools 2.42.3 and 2.48.0.

Steps to reproduce:
1. Run Flutter app in debug mode (VS Code or Android Studio).
2. Open Dart DevTools → Network.
3. Do some network requests → they are captured correctly.
4. Press "Clear".
5. Restart the app from IDE.
6. No new requests are captured anymore.

Expected result:
Network page continues to capture requests after Clear + Restart.

Actual result:
Network page stays empty, no requests appear.

**Environment:**
Flutter 3.35.4 • Dart 3.9.2 • DevTools 2.48.0
OS: Windows 10
IDE: VS Code (latest)

**Works fine with:**
Flutter 3.29.3 • Dart 3.7.2 • DevTools 2.42.3

首先，在 DevTools 2.48.0 上执行 Clear 后重启 IDE，以复现 Network 页面行为，然后与 2.42.3 进行比较，以缩小回归范围。跟踪 Network 页面在清除和重启过程中的请求捕获生命周期；当请求在重启后再次出现，并且该回归由测试覆盖时，即视为完成。

由索引模型根据 Issue 内容生成。