Search backend: remove `run` package

Created by: camdencheek

This moves the remaining stragglers left in the run package and deletes the package. Following up on a commitment I made to @rvantonder a very long time ago (I didn't forget, I just vastly underestimated how much work it would be to tease all of this apart).

For context, run was used as a bit of a dumping ground for everything that I just wanted to get out of graphqlbackend but didn't want to spend time teasing apart circular dependencies. It has persisted, but now it is gone.

Stacked on https://github.com/sourcegraph/sourcegraph/pull/39533

Test plan

Semantics-preserving, just a reorg.