Is there a point to unit tests that stub and mock everything public?

Question

When doing unit tests the "proper" way, i.e. stubbing every public call and return preset values or mocks, I feel like I'm not actually testing anything. I'm literally looking at my code and creating examples based on the flow of logic through my public methods. And every time the implementation changes, I have to go and change those tests, again, not really feeling that I'm accomplishing anything useful (be it mid- or long-term). I also do integration tests (including non-happy-paths) and I don't really mind the increased testing times. With those, I feel like I'm actually testing for regressions, because they have caught multiple, while all that unit tests do is show me that the implementation of my public method changed, which I already know.

Unit testing is a vast topic, and I feel like I'm the one not understanding something here. What's the decisive advantage of unit testing vs integration testing (excluding the time overhead)?