Fix tracing tests (#8022)

* Fix tracing tests

The tests were not working properly.

1. Some test was setting a global subscriber, this could lead to racy
conditions with other tests.

2. A logging test called `process::exit` which is completly wrong.

* Update client/tracing/src/lib.rs

Co-authored-by: David <[email protected]>

* Review comments

Co-authored-by: David <[email protected]>