After going through the test drives and the stupid dance with a couple of dealerships, the process smoothed out.
I checked out a few regional dealers, and called 3 different ones for prices. 2 of them were willing to me prices over the phone. Great. I found another regional dealer which had this "amazon-style" process. I could configure the car I wanted on their website, and it showed me the price. It was the best of the 3 prices I was quoted, so I "placed my order".
I got an auto-responder email email saying that paperwork was on the way and salesperson would be in touch. The guy called to discuss the trade in process, and tell me how the paperwork would be handled. In about 3 days I had a new car, delivered to my house, having never set foot in a dealership. They even did the trade in estimate remotely (I took some photos and sent them in).
Hopefully that's the trend things move in the future. It was a breeze.