Repo aware monitors: inline `searchClient.Execute`

Created by: camdencheek

This inlines searchClient.Execute so that we can mutate the job before running it. Moving forward, I think it makes sense for the Plan step to return a job directly, but we're not quite there yet, so this feels like a good compromise in the mean time.

Stacked on #32521 Pulled from #32464

Test plan

Semantics-preserving, manually tested.