Why would the “dirty play” be missed in a properly run 2-man system? ...
With the 2 man system there tends to be an "eyeball" hole behind (not in front of) the attack. Since you don't have that AR watching the 2nd to last defender, but the "Train" Ref watching is 15-20 yards behind the play (which means the Train Ref is sometimes 20-30 yards in front of the attacking teams defenders and the defending teams forwards (who are all walking/jogging back into position after a thwarted attack ), the opportunity for dirty forwards or defenders to give a little elbow, push or punch to each other or the GK that won't be seen by the Train Ref or the Lead Ref on the other side of the field exists.
With the 3 man system the extra pair of eyeballs that the trailing AR has helps keep the non-sense to a minimum.
To compound the above, when it comes to non-Varsity games there is a propensity to just pay for a single referee and the dirty play/nonsense increases.