[release-18.0] Fix flakiness in TestListenerShutdown (#17024)

[vitess-bot]

Description

This is a backport of #17024

[vitess-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 17186 -R vitessio/vitess
git reset --hard origin/release-18.0
git cherry-pick -m 1 2d1e96ac46641d660ee65e43ba69a156495e7b19

[vitess-bot]

Review Checklist

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

General

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