[release-21.0] Fix deadlock in messager and health streamer (#17230)

[vitess-bot[bot]]

Description

This is a backport of #17230

[vitess-bot[bot]]

Review Checklist

Hello reviewers! :wave: 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.

New flags

• [ ] Is this flag really necessary?
• [ ] Flag names must be clear and intuitive, use dashes (-), and have a clear help text.

If a workflow is added or modified:

• [ ] Each item in Jobs should be named in order to mark it as required.
• [ ] If the workflow needs to be marked as required, the maintainer team must be notified.

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[bot]]

Codecov Report

Attention: Patch coverage is 89.18919% with 4 lines in your changes missing coverage. Please review.

Project coverage is 67.51%. Comparing base (29f3980) to head (8c6d94e). Report is 1 commits behind head on release-21.0.

Files with missing lines	Patch %	Lines
go/vt/vttablet/tabletserver/health_streamer.go	82.60%	4 Missing :warning:

Additional details and impacted files

```diff @@ Coverage Diff @@ ## release-21.0 #17235 +/- ## ============================================= Coverage 67.50% 67.51% ============================================= Files 1570 1570 Lines 251375 251384 +9 ============================================= + Hits 169696 169722 +26 + Misses 81679 81662 -17 ```

:umbrella: View full report in Codecov by Sentry.
:loudspeaker: Have feedback on the report? Share it here.