Search UI: Decouple logic in `<SearchFilterSection />` component

Created by: vovakulikov

Follow-up for https://github.com/sourcegraph/sourcegraph/pull/40381

Background

In https://github.com/sourcegraph/sourcegraph/pull/40381 we decoupled the search bar panel and search filters section components. This allows reuse in different places without knowing a lot about their implementation. But FilterSection UI still has a lot of complexity, and it's hard to maintain.

This issue implies rewriting this component in a way where we would have a list of small and easy-to-maintain components for building another search UI filter section.

/cc @benvenker @lguychard