If you look for edge cases in integration tests, you will have a combinatorial explosion of integration tests and you will be adding much more work. Unit tests save time, not lose it.
I make this part of my filtering potential companies to work with now. I can't believe how often people avoid doing unit tests.
That's funny, since I wouldn't code at a place that mandates unit tests. Sure, they have a very minor role, in very specific cases, but I'd say 90% of projects can get 90% of the benefits by writing only integration tests with 90% coverage. If you'd like a more in-depth discussion of why integration testing is better: https://kentcdodds.com/blog/write-tests