backup_control.rs: remove unsafe & Mutex

[oech3]

maybe, closes #12240

[github-actions]

GNU testsuite comparison:

Skip an intermittent issue tests/date/date-locale-hour (fails in this run but passes in the 'main' branch)
Skip an intermittent issue tests/rm/isatty (fails in this run but passes in the 'main' branch)
Skip an intermittent issue tests/tail/tail-n0f (fails in this run but passes in the 'main' branch)
Skipping an intermittent issue tests/pr/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/printf/printf-surprise is now being skipped but was previously passing.

[codspeed-hq]

Merging this PR will not alter performance

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

Warning

Please fix the performance issues or acknowledge them on CodSpeed.

Performance Changes

	Mode	Benchmark	BASE	HEAD	Efficiency
❌	Memory	cp_recursive_deep_tree[(120, 4)]	562.5 KB	615.6 KB	-8.62%
⚡	Simulation	ls_recursive_balanced_tree[(6, 4, 15)]	52.3 ms	49.6 ms	+5.49%
⚡	Simulation	ls_recursive_wide_tree[(10000, 1000)]	34.3 ms	33 ms	+3.96%

Tip

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

---

_{Comparing oech3:unsafe-mutex (0e64be6) with main (0f538b9)²}

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. ↩

2. No successful run was found on main (3176027) during the generation of this report, so 0f538b9 was used instead as the comparison base. There might be some changes unrelated to this pull request in this report. ↩

[oech3]

Is this OK to merge?