I like the idea.
Maybe out of 100 servers, 20 or so are designated to solo mode for those who want to play more competitively. However, there's two main problems I can see with this.
Regulating and enforcing the no-teaming rule would be hard, as there won't be a moderator in every game and despite most people wanting to play solo in these games, there's always those who think it'd be funny to ruin it for everyone else.
On the topic of people who want to ruin the game, hackers would be even more advantaged as they won't face any teams, and hackers normally win 1 v 1s so they'd have a high chance of winning. Hopefully the crack down on hackers will make this less of a problem.