gh-37750: CI Build&Test: Fix "test modularized distributions" after #…

…37022

    
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- Cherry-picked updates of testing infrastructure from #35095.
- Add upper version bounds to avoid a regression in 4.14.1
- Fixes #37734.
- Also removes some unnecessary runs of the "CI Linux Incremental"
workflow.

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### ⌛ Dependencies

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URL: #37750
Reported by: Matthias Köppe
Reviewer(s): Kwankyu Lee