The random version exists because nobody wants people to spam the same champion all the time.
Honestly no version of URF is fun imo, if you get to pick your own champion it can either be too easy that you're not even trying (well that's how i feel) or the enemy team will just curb stomp you and your team with their own picks.
Having random picks doesn't help since people just call duo top or duo bot and will never agree to switch lanes if your random champion is against someone who seriously out snowballs you.
I know this mode is ment to be played for fun but imo it isn't that much fun.
I'd get a kick out of playing sona, wukong, karthus, nautilus, thresh, but after one game or two I get really sick of either version of URF.
Little off topic but I just wish Riot would allow players to play other maps in private games.