Cooperation isn't part of the rules, nor is it assumed.
I'm not sure where you're getting that idea. You adventure in a party. Solo combat isn't part of the way the rules have been balanced -- a party of four is.
The FR alone has far too many examples of human and demihuman types not cooperating. Working together is just one style of play, among what should be a large amount of equally balanced choices.
Groups working against each other is different from not being a cooperative game. In D&D, you play a party that goes against challenges, not against each other. If you want to go against each other, D&D isn't going to help you.
Low magic isn't balanced with high magic, vs. play isn't balanced with cooperative play. This is even true in MMORPG's like World of Warcraft: classes that work really well in a party environment (like the WoW Hunter or WoW Warrior) suffer HUGELY when arrayed against other players. If you want to play AGAINST other people, a different game is needed, since D&D's tools for that are pretty unapologetically awful. They aren't designed to be one lone guy against one lone guy and be good. That's what the minis game is for, or what CCGs are for. Not D&D.
It just seems wrong to me to assume that people in the world of D&D are somehow built differently so that they tend to agree more, especially when even default D&D has as many evil gods as it does good.
This is because the Evil is the Bad Guys and the Good is the Heroes, and D&D is designed with the idea that PC's are the heroes. The PC's won't be fighting each other, they'll be working together against the bad guys who are also working together. PC's not working well together is a problem because the game is not meant to handle that kind of environment.
If they weren't, you'd have more than two classes with access to healing magic, broader selection in weapon skills, more diversity in powers within a single class....a different sort of balance than you have when you assume a party atmosphere. And there's no reason that D&D should cater to vs. play, as far as I can see. There are other things that do that just fine.
All of the statements about cooperation being the name of the game make some pretty hefty assumptions about human nature that I'm finding hard to swallow.
This isn't about human nature. It's about the game's rules. It's not competative play, one person against another struggling for dominace where they will oppose each other individually. It's a game of heroic adventure, where a party of dedicated heroes fight and overcome challenges. That's not an assumption about human nature, that's an assumption about how the game is played. If you want to play it a different way (such as competitive play) be prepared to find holes in it. The only way competitive play works is party-based competitive play (or Gestalt characters, which are pretty much one-person parties.

). Otherwise, you're just trying to make the game do something the game was never meant to do, and saying it doesn't live up to your expectations.