Fix flakiness of ControllerPeriodicTasksIntegrationTest

[Jackie-Jiang]

Fix #13263

• Extend the wait time to reduce flakiness
• Use the new config keys instead of the deprecated keys (deprecated keys are tested within ControllerConfTest)
• Extract common gauge value comparison logic to improve readability

[codecov-commenter]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 62.06%. Comparing base (59551e4) to head (0f2c00d).
Report is 576 commits behind head on master.

Additional details and impacted files

@@             Coverage Diff              @@
##             master   #13337      +/-   ##
============================================
+ Coverage     61.75%   62.06%   +0.31%     
+ Complexity      207      198       -9     
============================================
  Files          2436     2545     +109     
  Lines        133233   139671    +6438     
  Branches      20636    21657    +1021     
============================================
+ Hits          82274    86691    +4417     
- Misses        44911    46431    +1520     
- Partials       6048     6549     +501     

Flag	Coverage Δ
custom-integration1	<0.01% <ø> (-0.01%)	⬇️
integration	<0.01% <ø> (-0.01%)	⬇️
integration1	<0.01% <ø> (-0.01%)	⬇️
integration2	0.00% <ø> (ø)
java-11	62.00% <ø> (+0.30%)	⬆️
java-21	61.95% <ø> (+0.32%)	⬆️
skip-bytebuffers-false	62.05% <ø> (+0.30%)	⬆️
skip-bytebuffers-true	61.89% <ø> (+34.16%)	⬆️
temurin	62.06% <ø> (+0.31%)	⬆️
unittests	62.06% <ø> (+0.31%)	⬆️
unittests1	46.59% <ø> (-0.30%)	⬇️
unittests2	27.76% <ø> (+0.03%)	⬆️

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