I've played random for a few months. As for pros and cons, the list of cons is definitely a lot bigger, but imo the pros still outweigh them.

The cons are pretty obvious, having to keep track of 9 matchups instead of 3, learning 3 races of hotkeys, ect. 
The pros are 1, your opponent won't know what to go for until he scouts (although it's pretty common for random players to announce what race they are playing at the start of the game, if they aren't interested in taking advantage of a gimmick to get ahead) and 2, and in my opinion the biggest reason to play random: You learn the ins and outs of all 3 races. MVP is a player that is known to play random almost exclusively during ladder for this exact reason. It's a great way to hyper develop your knowledge of all 3 races and help you play against them.

For example, any protoss can tell you that you will have 3 pylons by the 4ish minute mark. This is something not many zergs and terrans know, unless they have played protoss at some point. This is a great indicator to whether or not you are being proxied. i.e. if you scout your protoss opponent and he only has 2 pylons in his main at the 4 minute mark, either he's really bad, or he's built his 3rd pylon elsewhere on the map.

People say that you're just holding yourself back by playing random, and I would agree, IF you're playing random as your main race. Random should be used as a training tool to improve yourself with your main race, not as a main race itself.