[release-21.0] Fix how we cancel the context in the builtin backup engine (#17285)

[vitess-bot]

Description

This is a backport of #17285

[vitess-bot]

Review Checklist

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[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 67.53%. Comparing base (644f4a5) to head (f31c28e).
Report is 2 commits behind head on release-21.0.

Additional details and impacted files

@@               Coverage Diff                @@
##           release-21.0   #17291      +/-   ##
================================================
+ Coverage         67.49%   67.53%   +0.03%     
================================================
  Files              1570     1570              
  Lines            251391   251403      +12     
================================================
+ Hits             169678   169781     +103     
+ Misses            81713    81622      -91     

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