Search backend: unembed SearchInputs from searchResolver
Created by: camdencheek
Because SearchInputs is embedded in searchResolver, it is difficult to
tell exactly where it is being used since find references doesn't show
the places its fields are accessed through the searchResolver
type.
This just makes it a named field rather than an embedded field. This will make it easier to reason about some of the upcoming changes I'd like to make.
Stacked on #31337
Test plan
Semantics preserving and covered by tests.