refactor: Replace RecursiveMutex with Mutex in Shutdown()

Summary:

Step by step, going to replace all of the RecursiveMutex instances with the Mutex ones throughout the code base :)

Not sure if it is possible in all cases though...

This one is a low-hanging fruit.

This is a backport of core#19180

Test Plan: ninja all check-all

Reviewers: #bitcoin_abc, majcosta

Reviewed By: #bitcoin_abc, majcosta

Differential Revision: https://reviews.bitcoinabc.org/D9279