When talking about playcentric readings of Mario Kart in class, my team started in on a discussion of split-screen providing players different worlds to play in. Each screen could have a different level of competition because the other players are almost indistinguishable from AI if you don't care enough. This lets the players who are less competitive forget about the competition while those who are fighting for first focus in on the other real people. In this model, shells and other items are main communication (outside of screen peeking). They enable you to bash your way to the top, almost always screwing other players over. UI in the standard racing mode encourages competition (as does the general mode of racing): you get your placement in the race front and center. It's a quick check of the total map to see who's behind you. Items appear front-and-center to let you know you can attack the other players. So even though the split screen nature of the game can let you imagine yourself in your own race, the element of competition is omnipresent. Splitting up the screens only allows the player to set their own level of competition.
However, once a race is finished, players get a chance to view the highlight reel. And, our group did something I don't normally do, we watched it. The highlight reel provides a place to scale back out of the competition. All the players get slammed into the same space, reveling in good shell throws and nice passes. The highlight reels bring players together into one space, one mindset and helps negate the competitive load. We were all happy for other people to show up on the reel and Mario Kart escaped its reputation of ruining friendships. Considering I played with my final group team, this is a major plus. The different POVs in Mario Kart help maintain a healthy balance of competition and cooperation.
It is interesting that you mentioned that the split screen reminds people of the "omnipresent competition", and provides different worlds for different players. It seems that the split screen feature of Mario Kart did a great job to improve the experience for two types of players. For people who would love to take it as a competition, the split screen gives direct insight into other player's progress, and save them the effort to peek other's screens. For the less competitive group, on the other hand, the split screen would let them see each other's performance. Either way, a silent communication is added to the gameplay.