Been giving this one some thought, and unfortunately the <expression> token is unlikely to help. Since the screen is updated every second, the <expression> token will be re-evaluated (it won't be cached because its value will most likely change), which would cause the cell in which you place it to be re-rendered, causing any video to restart. So even if we could come up with a way to have the <expression> token provide a random way to specify a video, it wouldn't allow the video to play because it would be continually restarting.
Are the videos of different lengths? Would just one of them be picked and remain on that screen until the tournament starts? Or would you want it to [randomly] cycle through the pre-start screens (each with a different video) until the tournament starts?