I have biked on both places (In more suburban areas). The roads tend to be terrifying (Because try as I might, I can't keep up with 30 mph traffic.) The sidewalks tend to be empty. I also didn't blow through intersections, which minimized the risk of getting hit in one.

Cars aren't the only thing with doors, but they are the only thing that will door you into traffic. Doors flush with buildings are also quite rare, and are quite rarely opened. (Except store doors - but you don't want to bike near them anyways, unless you want to run into a sign, or a table, or an outdoor display.)

I feel like I'm missing something. I can't understand how it's more likely for a car to hit a cyclist on the sidewalk than the road. Cars drive in the road. They do not drive on the sidewalk.

I hit a car while riding on the sidewalk as it was exiting an alley. The problem is that when you are riding on the sidewalk in a city, the corners are too blind. In order to guarantee avoiding an accident you essentially need to stop at each driveway and alley. Most cyclists and drivers aren’t going to do that. On the road you are far enough back that you can see cars exiting; they can see you as well.

They drive across the sidewalk when entering or leaving driveways, and drivers do not always see cyclists in those circumstances.