SQLite is not <your actual rdbms>... it's SQLite. A database that does not enforce column types and lacks most of the advanced features of a real rdbms. Using SQLite will work great right up until the point where it won't and you get your fingers burnt.
If you separate your concerns properly you won't need to mock the database layer either. Mocking is just one part of the trifecta of good testing, along with Stubbing and Faking.
For most things it would make more sense to fake the database layer or stub the database layer in your "logic" layer.
However, if your application makes heavy use of the rdbms then you should test that layer too: In your integration tests and not your unit tests. Most places that interact with an RDBMS treat them like black boxes and not like a business layer of its own. You really need integration tests to ensure things like constraints and your business rules are captured properly... most people never bother.
The real problem with TDD and its methodologies isn't TDD itself; it's people shoehorning about 10% of what proper testing should be into two narrow groups: stuff that you can do with "unit tests" and "things we can mock." There's a lot more to it than just those two things.
If you separate your concerns properly you won't need to mock the database layer either. Mocking is just one part of the trifecta of good testing, along with Stubbing and Faking.
For most things it would make more sense to fake the database layer or stub the database layer in your "logic" layer.
However, if your application makes heavy use of the rdbms then you should test that layer too: In your integration tests and not your unit tests. Most places that interact with an RDBMS treat them like black boxes and not like a business layer of its own. You really need integration tests to ensure things like constraints and your business rules are captured properly... most people never bother.
The real problem with TDD and its methodologies isn't TDD itself; it's people shoehorning about 10% of what proper testing should be into two narrow groups: stuff that you can do with "unit tests" and "things we can mock." There's a lot more to it than just those two things.