Unquarantine QA for running pipelines with variables

What does this MR do and why?

For #375552 (closed)

When we migrated the Run Pipeline form to GraphQL, these specs were quarantined because they suddenly became flaky. The issue was that the migration did not implement polling the backend cache for updated values, which was in the original implementation. A lot of workarounds for QA was added because of this.

With the fix in #381233 (closed) we can unquarantine the tests again and remove these unnecessary workarounds (like page.refreshing to access the new cache values).

See #381233 (closed) for a more in-depth discussion of the issue.

Screenshots or screen recordings

N/A

How to set up and validate locally

Pipeline should pass ✅

MR acceptance checklist

This checklist encourages us to confirm any changes have been analyzed to reduce risks in quality, performance, reliability, security, and maintainability.

• I have evaluated the MR acceptance checklist for this MR.