[release-19.0 backport] Mysqld: capture mysqlbinlog std error output (#15278)

[shlomi-noach]

release-19.0 backport of #15278

Backporting manually and a bit late, following request by the community.

Coordinated with @systay , the release owner for 19.0.

[vitess-bot]

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[codecov]

Codecov Report

Attention: Patch coverage is 0% with 11 lines in your changes are missing coverage. Please review.

Project coverage is 67.42%. Comparing base (b2291f6) to head (3581155).
Report is 1 commits behind head on release-19.0.

Files	Patch %	Lines
go/vt/mysqlctl/mysqld.go	0.00%	11 Missing ⚠️

Additional details and impacted files

@@               Coverage Diff                @@
##           release-19.0   #15379      +/-   ##
================================================
- Coverage         67.42%   67.42%   -0.01%     
================================================
  Files              1560     1560              
  Lines            192390   192405      +15     
================================================
+ Hits             129718   129723       +5     
- Misses            62672    62682      +10     

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