I think that approach would work for a typical developer of a typical web app. But not all places are the same. You can't easily test many things. You can have correct tests that miss some cases. You can have incorrect tests matching incorrect code. You can have code fixing the result rather than the cause.

Basically unless you trust developers to never be wrong, I'd do reviews. And then if you do trust them, I'd still do reviews to make sure everyone learns what's happening outside of their area and where common utilities can prevent code duplication.

In short - there are things that cannot be realistically automated or simplified into compilation checks. If they don't occur in your environment - great. When they do, look again at code reviews.