[release-20.0] Fix deadlock in messager and health streamer (#17230)

[vitess-bot]

Description

This is a backport of #17230

[vitess-bot]

Hello @GuptaManan100, there are conflicts in this backport.

Please address them in order to merge this Pull Request. You can execute the snippet below to reset your branch and resolve the conflict manually.

Make sure you replace origin by the name of the vitessio/vitess remote

git fetch --all
gh pr checkout 17234 -R vitessio/vitess
git reset --hard origin/release-20.0
git cherry-pick -m 1 9fdfdb962a6f38b61dc171d0121371ae7be075a0

[vitess-bot]

Review Checklist

Hello reviewers! 👋 Please follow this checklist when reviewing this Pull Request.

General

• Ensure that the Pull Request has a descriptive title.
• Ensure there is a link to an issue (except for internal cleanup and flaky test fixes), new features should have an RFC that documents use cases and test cases.

Tests

• Bug fixes should have at least one unit or end-to-end test, enhancement and new features should have a sufficient number of tests.

Documentation

• Apply the release notes (needs details) label if users need to know about this change.
• New features should be documented.
• There should be some code comments as to why things are implemented the way they are.
• There should be a comment at the top of each new or modified test to explain what the test does.

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If a workflow is added or modified:

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Backward compatibility

• Protobuf changes should be wire-compatible.
• Changes to _vt tables and RPCs need to be backward compatible.
• RPC changes should be compatible with vitess-operator
• If a flag is removed, then it should also be removed from vitess-operator and arewefastyet, if used there.
• vtctl command output order should be stable and awk-able.

[codecov]

Codecov Report

Attention: Patch coverage is 94.28571% with 2 lines in your changes missing coverage. Please review.

Project coverage is 66.42%. Comparing base (810ef09) to head (abda457).
Report is 1 commits behind head on release-20.0.

Files with missing lines	Patch %	Lines
go/vt/vttablet/tabletserver/health_streamer.go	90.47%	2 Missing ⚠️

Additional details and impacted files

@@               Coverage Diff                @@
##           release-20.0   #17234      +/-   ##
================================================
+ Coverage         66.39%   66.42%   +0.02%     
================================================
  Files              1541     1541              
  Lines            244065   244074       +9     
================================================
+ Hits             162044   162118      +74     
+ Misses            82021    81956      -65     

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