[release-20.0] Fix flakiness in `TestListenerShutdown` (#17024)

[vitess-bot[bot]]

Description

This is a backport of #17024

[vitess-bot[bot]]

Hello @frouioui, there are conflicts in this backport.

Please address them in order to merge this Pull Request. You can execute the snippet below to reset your branch and resolve the conflict manually.

Make sure you replace origin by the name of the vitessio/vitess remote

git fetch --all
gh pr checkout 17188 -R vitessio/vitess
git reset --hard origin/release-20.0
git cherry-pick -m 1 2d1e96ac46641d660ee65e43ba69a156495e7b19

[vitess-bot[bot]]

Review Checklist

Hello reviewers! :wave: Please follow this checklist when reviewing this Pull Request.

General

• [ ] Ensure that the Pull Request has a descriptive title.
• [ ] Ensure there is a link to an issue (except for internal cleanup and flaky test fixes), new features should have an RFC that documents use cases and test cases.

Tests

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Documentation

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• [ ] There should be a comment at the top of each new or modified test to explain what the test does.

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