backup_control.rs: remove unsafe & Mutex

[oech3]

maybe, closes #12240

[github-actions]

GNU testsuite comparison:

Skip an intermittent issue tests/misc/tty-eof (fails in this run but passes in the 'main' branch)
Skip an intermittent issue tests/pr/bounded-memory (fails in this run but passes in the 'main' branch)
Congrats! The gnu test tests/expand/bounded-memory is now passing!
Congrats! The gnu test tests/rm/many-dir-entries-vs-OOM is now passing!

[codspeed-hq]

Merging this PR will not alter performance

✅ 132 untouched benchmarks
⏩ 253 skipped benchmarks¹

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_{Comparing oech3:unsafe-mutex (ebdd879) with main (ef50d8b)}

Footnotes

1. 253 benchmarks were skipped, so the baseline results were used instead. If they were deleted from the codebase, click here and archive them to remove them from the performance reports. ↩

[oech3]

Is this OK to merge?

[oech3]

@cakebaker Would you review this?