[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

Start by reproducing the Network page behavior with Clear followed by an IDE restart on DevTools 2.48.0, then compare it with 2.42.3 to narrow the regression. Trace the Network page's request-capture lifecycle around clearing and restarting; done means requests appear again after the restart and the regression is covered by a test.

Written by the indexing model from the issue text.