Rewrite _many_ tests to use vtctldclient invocations, mostly non-output related stuff

[ajm188]

Description

I arbitrarily picked this as a stopping point for this batch. I'll do another round after this!

I'm focusing on the non-output-focused stuff for now to put off dealing with minor changes in output that tests may depend on, but those are coming up.

Related Issue(s)

#15273

Checklist

• "Backport to:" labels have been added if this change should be back-ported to release branches
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• Tests were added or are not required
• Did the new or modified tests pass consistently locally and on CI?
• Documentation was added or is not required

Deployment Notes

[vitess-bot]

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[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Comparison is base (54c6dfc) 67.44% compared to head (7c538ab) 67.45%.

Additional details and impacted files

@@           Coverage Diff           @@
##             main   #15270   +/-   ##
=======================================
  Coverage   67.44%   67.45%           
=======================================
  Files        1561     1561           
  Lines      193193   193193           
=======================================
+ Hits       130304   130314   +10     
+ Misses      62889    62879   -10     

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[shlomi-noach]

Nice!