This is a reland of 138bcfc3

Fixed all the data races and ran TSAN locally to confirm.

Original change's description:
> [cpu-profiler] Add more logging to find flaky failure
>
> There is a flaky 5x failure in the tree which I can't reproduce locally.
> This extra logging will help flush out what the problem is.
>
> Bug: v8:8649
>
> Change-Id: If36d2ce0f4feb398d7d746d69b417bb55a714422
> Reviewed-on: https://chromium-review.googlesource.com/c/1402787
> Commit-Queue: Peter Marshall <petermarshall@chromium.org>
> Reviewed-by: Camillo Bruni <cbruni@chromium.org>
> Cr-Commit-Position: refs/heads/master@{#58796}

Bug: v8:8649
Change-Id: I53e293ef85a54d4b2b39aa3b980832031201aa0c
Reviewed-on: https://chromium-review.googlesource.com/c/1411633
Commit-Queue: Peter Marshall <petermarshall@chromium.org>
Reviewed-by: Jakob Gruber <jgruber@chromium.org>
Cr-Commit-Position: refs/heads/master@{#58833}