From my experience trusting a unit test instead of looking at things in the browser is a certain recipe for introducing bugs. Not sure if this is what you are advocating or if I misunderstood.
In the browser, testing edge cases sometimes requires custom headers, encoding of data to create authorization tokens, etc. This means you have to either have amazing browser tools (which I've yet to find) or use a combination of browser and shell to achieve what I need.
In TDD, most setup can be automated with simple function calls. In addition, well made frameworks, such as PHP's Symfony, have utilities that even avoid making real HTTP requests, so tests run faster than using a browser, but the result is the same.
I'm not saying everyone should do TDD, but from my experience, it can lead to increased productivity and fewer bugs in some cases.
By the way, if you know of browser tools that make testing easier, I'd be glad to learn more. I use TDD because I lack the browser tools. If I had the right tools, maybe I'd consider going back to in browser testing.