[8.12] Revert "Revise wait_for_integration ES implementation (#12150)" (backport #12240)

[mergify]

This is an automatic backport of pull request #12240 done by Mergify.
Cherry-pick of 54e367c has failed:

On branch mergify/bp/8.12/pr-12240
Your branch is up to date with 'origin/8.12'.

You are currently cherry-picking commit 54e367c5b.
  (fix conflicts and run "git cherry-pick --continue")
  (use "git cherry-pick --skip" to skip this patch)
  (use "git cherry-pick --abort" to cancel the cherry-pick operation)

Changes to be committed:
	modified:   internal/beater/beater.go
	modified:   internal/beater/beatertest/server.go
	modified:   internal/beater/server_test.go
	modified:   internal/beater/waitready.go

Unmerged paths:
  (use "git add <file>..." to mark resolution)
	both modified:   internal/beater/checkintegration.go

To fix up this pull request, you can check it out locally. See documentation: https://docs.github.com/en/github/collaborating-with-pull-requests/reviewing-changes-in-pull-requests/checking-out-pull-requests-locally

---

Mergify commands and options

More conditions and actions can be found in the documentation.

You can also trigger Mergify actions by commenting on this pull request:

• @Mergifyio refresh will re-evaluate the rules
• @Mergifyio rebase will rebase this PR on its base branch
• @Mergifyio update will merge the base branch into this PR
• @Mergifyio backport <destination> will backport this PR on <destination> branch

Additionally, on Mergify dashboard you can:

• look at your merge queues
• generate the Mergify configuration with the config editor.

Finally, you can contact us on https://mergify.com

[axw]

Closes #12239