Merge #19180: refactor: Replace RecursiveMutex with Mutex in Shutdown()

1a9ef1d398dd14728b6bc67a89139cdf827c9753 refactor: Replace RecursiveMutex with Mutex in Shutdown() (Hennadii Stepanov)

Pull request description:

  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.

ACKs for top commit:
  MarcoFalke:
    ACK 1a9ef1d398dd14728b6bc67a89139cdf827c9753 Shutdown is not recursive, so the same thread can never lock twice (UB)
  vasild:
    ACK 1a9ef1d3 verified manually that `Shutdown()` is not called from places that could be called from inside `Shutdown()`.

Tree-SHA512: 362a507b1a6f97dc351f708224aedbfe4bee03c4398f394d78ee31c24d76a7012ffff0e6766866cd5fd9a8e0d8840f05a2741111fe583aa20d45f0af3df0dcfa

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