Standalone Profiler doesn't record data when the Target is changed to and from Build

Steps to reproduce:

1. Create and open a new project
2. In the Player Settings > Resolution and Presentation enable "Run In Background"
3. Build and run the project with Development Mode on
4. Open the Window > Analysis > Profiler (Standalone)
5. Connect the Profiler to the build
6. Observe the Profiler

Actual results: Profiler doesn't start recording data

Expected results: Profiler starts recording data immediately

Reproducible with versions: 2023.1.0a10, 6000.0.73f1, 6000.3.13f1, 6000.4.3f1, 6000.5.0b4, 6000.6.0a3

Tested on (OS): Windows 11 Pro (25H2)

Workaround: Disable and re-enable Profiler recording

Notes:

• Reproducible every time the profiler is switched to Build or from Build
• Does not reproduce if the profiler is already running and it is switched from Editor -> Play Mode, or Play Mode -> Editor
• When switching from a recorded Build to Edit Mode (or a running Play Mode), an "Unknown profiler sampler in thread block:" error will be thrown
• Doesn't reproduce with the embedded Profiler