[BACKPORT] Fix intermittent failure in ApiKeyIntegTests (#38627)

[bizybot]

Few tests failed intermittently and most of the
times due to invalidated or expired keys that were
deleted were still reported in search results.
This commit removes the test and adds enhancements
to other tests testing different scenario's.

When ExpiredApiKeysRemover is triggered, the tests
did not await its termination thereby sometimes
the results would be wrong for a search operation.

DELETE_INTERVAL setting has been further reduced to
100ms so we can trigger ExpiredApiKeysRemover faster.

Closes #38408

• Have you signed the contributor license agreement?
• Have you followed the contributor guidelines?
• If submitting code, have you built your formula locally prior to submission with gradle check?
• If submitting code, is your pull request against master? Unless there is a good reason otherwise, we prefer pull requests against master and will backport as needed.
• If submitting code, have you checked that your submission is for an OS that we support?
• If you are submitting this code for a class then read our policy for that.

[elasticmachine]

Pinging @elastic/es-security