Integration tests tell you whether it’s working. Unit tests tell you what isn’t working. So long as everything is working, you “don’t need” the unit tests – but once something is wrong, it’s very nice to have the unit test point you directly to the problem. As you say, they serve different purposes; it’s good to have both.
To directly address your subject: integration tests aren’t a problem, aren’t the problem. Using them instead of unit tests is.