[release-21.0] Fix potential deadlock in health streamer (#17261)

[vitess-bot]

Description

This is a backport of #17261

[vitess-bot]

Review Checklist

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General

• Ensure that the Pull Request has a descriptive title.
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Backward compatibility

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[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 67.53%. Comparing base (e72904b) to head (ca974e2).
Report is 1 commits behind head on release-21.0.

Additional details and impacted files

@@               Coverage Diff                @@
##           release-21.0   #17270      +/-   ##
================================================
+ Coverage         67.50%   67.53%   +0.02%     
================================================
  Files              1570     1570              
  Lines            251389   251391       +2     
================================================
+ Hits             169695   169765      +70     
+ Misses            81694    81626      -68     

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