[chore][service/internal/proctelemetry] Remove flaky goleak test (#9231)

**Description:** <Describe what has changed.>
<!--Ex. Fixing a bug - Describe the bug and how this fixes the issue.
Ex. Adding a feature - Explain what this achieves.-->
Remove the goleak test from this package for now as it's flaky. This is
reverting some of #9227, but leaving some added `Shutdown` calls.

**Link to tracking Issue:** <Issue number if applicable>
#9230: This issue will remain open to track debugging and re-enabling
the test

Co-authored-by: Bogdan Drutu <[email protected]>