By faking the DB I meant either running a local, prefilled fake DB server for every test or faking the interface to the DB.

Which one you should do depends on how complex your interactions with the DB are.

Some apps (e.g. CRUD) have half of their business logic encoded in DB queries in which case faking the calls is a bad idea.

Others only do, like, 2 simple queries. In this case there's no point running an actual database outside of a couple of E2E tests.