Search backend: simplify client and package structure
Created by: camdencheek
Without the expand step removed in the https://github.com/sourcegraph/sourcegraph/pull/39520, we've broken some circular dependency issues that have forced our search client and package structure to be way more complex than needed. This PR starts unraveling some of that.
Notably, this PR removes the execute
package and significantly slims down the run
package, which I plan to eliminate altogether in a later PR.
Each of these commits is small and self-contained and has a message describing what it does, but they felt best to be reviewed all together.
Test plan
Semantics-preserving. Just moving things around.