[GaryVoshol]

The problem really is too much randomness, not too little. What we really want is a defined tournament scheme, not a random scheme. Total randomness can create some surprisingly non-random looking results.

There are very well defined schemes for tournaments involving 4 teams/individuals. Bridge, for example. Given "X" players (x divisible by 4), a schedule can be set up so that you are partnered with each other player once, and against every other player twice. (Since at any one time, you have one partner and 2 opponents.) Or for round-robin leagues, where every team plays every other team once.

I don't know if such schemes are defined for a 3v3 competition.

The restrictions on not playing too many matches within a given time period also puts constraints on the selection process.