Except that wont increase your chances of getting a game quicker.
Actually, you're wrong.
In beta, when we could read a particular game file, we could see that the longer you search for a game, the higher the elo spread between you and your opponent will possibly be.
Let's say you are elo 1000.
You start searching for a game as random, automatch will search for a german or soviet opponent between 950 and 1050.
Then 1 minute later, it will search for a german or soviet player between 900 and 1100.
Then 1 minute later, between 850 and 1150.
etc etc.
Now, without a random function, if you search as a 1000 elo german player and don't find a game, but with an other german player searching too with 1070 elo in the automatch pool. After x time you decide to restart the search as a soviet player, you'll have to wait at least 1 minute to be able to be matched with him.
While with a random faction button, you wouldn't have had to wait. You would have met him way sooner, 1 minute after starting your search.