Sure, integration tests "save" you from writing pesky unit tests, and changing them frequently after every refactor.
But how do you quickly locate the reason that integration test failed? There could be hundreds of moving parts involved, and any one of them malfunctioning, or any unexpected interaction between them, could cause it to fail. The error itself would likely not be clear enough, if it's covered by layers of indirection.
Unit tests give you that ability. If written correctly, they should be the first to fail (which is a good thing!), and if an integration test fails, it should ideally also be accompanied by at least one unit test failure. This way it immediately pinpoints the root cause.
The higher up the stack you test, the harder it is to debug. With E2E tests you're essentially debugging the entire system, which is why we don't exclusively write E2E tests, even though they're very useful.
To me the traditional test pyramid is still the best way to think about tests. Tests shouldn't be an afterthought or a chore. Maintaining a comprehensive and effective test suite takes as much hard work as, if not more than, maintaining the application itself, and it should test all layers of the system. But if you do have that, it gives you superpowers to safely and reliably work on any part of the system.