[native] Use useKeyserverRecoveryLogIn from SQLiteDataHandler

Summary:
This unifies the logic with KeyserverConnectionHandler, and reduces code duplication.

Depends on D11226

Test Plan: Part 4.6 of this test plan: https://gist.github.com/Ashoat/75ab690d5c53cdd68a51b02e03e27c58

Reviewers: inka, tomek

Reviewed By: tomek

Differential Revision: https://phab.comm.dev/D11227