Unify log queries to only have SQLX code in the runtime and use it for both the log client and for tests
<img width="883" height="84" alt="Screenshot 2026-01-29 at 11 13 12" src="https://github.com/user-attachments/assets/090a2fec-94ed-4c0f-aee5-1653ed8b1439" />
Add a log DB. The goal is just to store our logs in a `.sqlite` DB to make it easier to crawl them and drop the oldest ones.