[release-19.0] Fix potential deadlock in health streamer (#17261)

[vitess-bot]

Description

This is a backport of #17261

[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.

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]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 65.16%. Comparing base (185b31e) to head (53ec6bb).
Report is 1 commits behind head on release-19.0.

Additional details and impacted files

@@               Coverage Diff                @@
##           release-19.0   #17268      +/-   ##
================================================
+ Coverage         65.14%   65.16%   +0.01%     
================================================
  Files              1560     1560              
  Lines            238303   238305       +2     
================================================
+ Hits             155251   155283      +32     
+ Misses            83052    83022      -30     

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

---

🚨 Try these New Features:

• Flaky Tests Detection - Detect and resolve failed and flaky tests
• JS Bundle Analysis - Avoid shipping oversized bundles