tee: merge logic of flush into write

[xtqqczze]

Closes #11914

[github-actions]

GNU testsuite comparison:

Skip an intermittent issue tests/date/resolution (fails in this run but passes in the 'main' branch)
Skipping an intermittent issue tests/tail/inotify-dir-recreate (passes in this run but fails in the 'main' branch)

[oech3]

Since tee.rs always call flush() after write_all(), we can just merge those functions.

[codspeed-hq]

Merging this PR will not alter performance

✅ 309 untouched benchmarks
⏩ 46 skipped benchmarks¹

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_{Comparing xtqqczze:tee-retain-with (c825f07) with main (34fd4be)}

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. ↩