Refactor symbols gitserver client and make sure we always use the DB connection
Created by: eseliger
Stacked on top of https://github.com/sourcegraph/sourcegraph/pull/39503. This also addresses a comment from Ryan here: https://github.com/sourcegraph/sourcegraph/pull/38635#issuecomment-1184105336. Review without whitespace changes.
Test plan
Test suite and main dry run.