To be clear, I don't think we disagree very much

I've seen people only run tests and break things (because the thing they broke wasn't covered by tests), I've seen people try to fix things and not verify that their fix works, etc

Good tests are sufficient in many cases to be confident that your code still works. But in general tests don't cover a lot of fundamental behavior, and if you don't exercise that fundamental behavior in one way or another, then you don't know that your code works