Stop running certain tests on master and stable branches

Summary

Context: #1406 (comment 1675813895)

Let's look into only running certain tests (like QA and backup/restore specs) on MR pipelines, and removing them from stable and master pipelines.

Running them there is fairly redundant as they have already been validated in the MR's merge results pipelines, and failures there are almost always transient - yet it takes significant team time to triage these failures and retry them until they're passing.

The goal here is to still run the tests, but run them strategically to ensure they give us proper and timely feedback without detracting unnecessarily from the team's time.

Acceptance criteria

• Identify an ideal workflow for where and when to run tests
• Create follow-up issues to implement this workflow in each of our relevant projects (Omnibus, Charts, Operator, etc.)