A unit test should only test a single class so that means you need to mock out all other classes which your class depends on. Mocking out dependencies in the test code is difficult or not feasible and that's why developers often resort to Dependency Injection in their source code but as mentioned before, it is an antipattern.
Also, I noticed that a lot of developers confuse unit tests with integration tests but the definition is quite clear: If your test covers more than one class without mocks, then it's an integration test. Integration testing does not necessarily mean end-to-end testing. It could just be a single class with its internal dependencies.
The only reason dependency injection gets a bad rep is magical frameworks which obscure the actual wiring and end up causing bigger problems.
I think that the class which ineteracts with the database directly via the client should be tightly coupled to the database client. It's not very often that you change database and when you do, you can just swap out that entire class completely. Classes which interact with the database should expose simple interfaces for performing actions against the database and those wrappers should be replaceable.