Hi @Mute64, welcome. You are able to run a random game via the controller, but it's not exactly obvious. On a 360 controller you can press Y to bring up the top menu, then use the arrows to choose items, and press A to select.
You might find Big Box Mode better suited for that, though, if you haven't played with it already. It's currently only available in the beta release, so you'll have to enable beta updates under Options. I do still need to add the random game feature to Big Box, though.