[release-20.0] 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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Tests

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Backward compatibility

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

Additional details and impacted files

@@               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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