Allright. After sleeping on it, I've come to some conclusions.
See, when I first wrote this story, it was designed for my ttrpg that I play with my friends irl. The problem is that they're the kind of players that actively rebel against the story line, so I had to write in certain precautions, like making it so that all the other machines were full. I realize, however, such precautions are not necessary. So, not all the machines are taken up, but it is busy. I will also be sure to mention that this is one of the larger and more expensive arcades, so the addition of such a large machine would be more plausible. By freeing up the other machines, it also gives the characters time to meet and greet a bit before they all sit down.
As for the odds of six strangers being in the same place at the same time and doing the same thing... it's an arcade. Almost everyone in there would be a stranger, and there'd be a lot more than six people. The reason each character decides to sit at that machine is dependent on the character.