I'm gonna go out on a limb here and say that a good chunk of competitive TF2 players do it out of interest and passion and that for them, playing 6's IS a big part of their social life. Not to mention a lot of the non-Invite players are between the ages of teens to mid-20's, where these people would be playing video games with friends to have fun for seven hours a day regardless if it was competitive. I can only assume it's why there are Main and IM teams out there who will scrim five days a week for the chance of winning $30.
I don't see a reason why TF2 players can't play on the weekends aside from the fact that Sat + Sun is usually when Highlander teams scrim, and 6's + HL players have made an invisible pact to not try to overlap each other as much as they can so a player is able to play on two teams.