Deduplicate testing dependencies by referencing [testing-integration]

[Avasam]

Summary of changes

This was my original idea, but in #4257 (comment), @abravalheri mentioned that:

[...] I think we just remove the testing-integration and use testing everywhere...

Although it might add a bit of overhead for the integration tests, it will simplify and streamline the setup... This way we reduce the divergence with skeleton. The additional overhead in the integration tests should not be too much, and the integration tests just run before the releases anyway.

But I'm still opening this PR as an option / for comparison.

Accepting this closes #4282

Pull Request Checklist

• Changes have tests (these are test changes)
• News fragment added in newsfragments/. (no user facing changes)
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