Follow-up of https://github.com/zed-industries/zed/pull/45365

* Stops printing backtraces for server diagnostics pulls on error

<img width="1728" height="1084" alt="backtrace"
src="https://github.com/user-attachments/assets/b2e75e73-611a-428b-8056-f0757e51adf2"
/>

* Reduce one `.detach` codepath into `.await`

Release Notes:

- N/A