[release-20.0] Fix potential deadlock in health streamer (#17261)

[vitess-bot[bot]]

Description

This is a backport of #17261

[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

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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

All modified and coverable lines are covered by tests :white_check_mark:

Project coverage is 66.41%. Comparing base (495a44a) to head (3a27494). Report is 2 commits behind head on release-20.0.

Additional details and impacted files

```diff @@ Coverage Diff @@ ## release-20.0 #17269 +/- ## ================================================ + Coverage 66.38% 66.41% +0.02% ================================================ Files 1541 1541 Lines 244074 244076 +2 ================================================ + Hits 162039 162098 +59 + Misses 82035 81978 -57 ```

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