No matter what user agent you use, Chrome won't show you how your page looks in a non-WebKit layout engine like Gecko, Trident, or Presto.

Different browsers (well, their rendering engine) will render html differently. The reason ie6 got so much slack years ago was because it didn't render 'correctly' at all. Web developers need to test in different browsers (/different versions) to check everyone is seeing the same thing.

overriding the UA just shows if the server serves different html to different clients

Nope; just today I ran into a bug where a click was being handled differently in an iPhone than it was in Chrome.