Being familiar with the competitive gaming culture around the game League of Legends, I have some basis for comparison when I watch King of Kong. In short, the competitive drama in King of Kong is when some people take something too seriously. And that is not a bad thing.
Having paid attention to what is basically the Major League of League of Legends, I know that what often excites people is the rivalries between the teams and players. In all honesty, much of the spectacular appeal in competitive events such as pro sports and competitive gaming lies in the people playing them, since the events themselves often do not have an interesting or existing narrative: watching Donkey Kong is watching a little man jumping over barrels and fireballs over and over, watching League of Legends is watching two teams of ten characters clobbering each other with fancy graphics, watching football (American or European) is watching a bunch of sweaty men/women going after a ball.
When the players generate drama between themselves, people pay attention. Famous biker is found to be doping? National television headlines the news. An American League of Legends player trash-talks another one on camera? Everyone posts the video on the front page of reddit. Drama between two Donkey Kong players? A documentary is made. These events draws attention from both the people who are familiar to the games and those who are not. I personally have not heard of Twin Galaxies and the competitive arcade culture until I saw the film. Drama generates excitement and exposure, and they in turn help legitimize competitive gaming.
Yes, I said “legitimize.” No, don’t pretend some of you don’t think gaming is dumb, with your jerseys and team paraphernalia. Joking aside, competitive gaming, much like competitive sports, puts the player or team on the spot, and often deemed the lesser if they are not successful or victorious. In situations like that, attitudes come into play, and people butt heads. Billy Mitchell probably believed he had to protect his position as the best player of Donkey Kong around, thus leading to some of his more unsavory comments. Perhaps Billy Mitchell is not the most likable person around, or perhaps he is a sore loser, but personalities like him create stories because of their undeniable skill set at their games and the drama they create.
FYI: Billy Mitchell, Steve Wiebe, and Walter Day are still involved in competitive gaming today. There are some videos of them playing in events on the Twin Galaxies website.