It would often be quite impossible to determine which team scored a goal, for instance in 'scrimmage' situations, or own goals.

Plus it would create weird situations where a attacking team would stop a goal from the other attacking team on the line and try to get in on their own name.

That's a problem in 2-team football, anyway (the individual who scored always has to be identified), although the point you raise about own goals is an interesting one. Maybe an own-goal should award half a goal to the other two teams?

The player who scored is identified, but it doesn't really matter because it goes to the scoring team regardless of who scored it.

Me too, the shirt rocks. Or please share the image in HQ!