How about a "golf" mode?
Perhaps you could develop a mode (I'd call it "golf") in which you get dropped into a location, perhaps with only rotate and zoom options, and then you get three chances to guess where you're at. After each guess, GeoGuessr would only report the distance you've guessed from the target location. Let's say you play 9 or 18 "holes" and record the total distance for your final guesses on each "hole". I think the game could be played quickly, and would allow new users to quickly familiarize themselves various locations, terrain types, etc.