backup_control.rs: remove unsafe & Mutex

[oech3]

maybe, closes #12240

[github-actions]

GNU testsuite comparison:

Skipping an intermittent issue tests/cut/bounded-memory (passes in this run but fails in the 'main' branch)
Skipping an intermittent issue tests/tail/retry (passes in this run but fails in the 'main' branch)
Note: The gnu test tests/dd/no-allocate is now being skipped but was previously passing.
Note: The gnu test tests/tail/tail-n0f is now being skipped but was previously passing.
Congrats! The gnu test tests/pr/bounded-memory is now passing!

[codspeed-hq]

Merging this PR will improve performance by 7.71%

⚠️ Different runtime environments detected

Some benchmarks with significant performance changes were compared across different runtime environments,
which may affect the accuracy of the results.

Open the report in CodSpeed to investigate

⚡ 2 improved benchmarks
❌ 1 regressed benchmark
✅ 314 untouched benchmarks
⏩ 46 skipped benchmarks¹

Warning

Please fix the performance issues or acknowledge them on CodSpeed.

Performance Changes

	Mode	Benchmark	BASE	HEAD	Efficiency
❌	Simulation	sort_long_line[10000]	423 µs	437.6 µs	-3.34%
⚡	Memory	cp_recursive_deep_tree[(120, 4)]	699.2 KB	562.5 KB	+24.31%
⚡	Simulation	ls_recursive_balanced_tree[(6, 4, 15)]	50.9 ms	48.9 ms	+4.01%

Tip

Investigate this regression by commenting @codspeedbot fix this regression on this PR, or directly use the CodSpeed MCP with your agent.

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_{Comparing oech3:unsafe-mutex (05f1beb) with main (d09f351)}

Footnotes

1. 46 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. ↩