I think the OP makes a valid point, yes I know the rules of golf allow for any club to be used on the green although I believe this is because in the old days balls were not lifted and marked with ball markers so if your opponent's ball was in direct line between your ball and the hole you were 'stymied' so you either had to hit around it or over it.
Most definitions of the word 'Putt' state "to strike (a golf ball) gently so as to make
it roll along the green into the hole"
And the Scottish origin of the word states "...literally, shove, gentle push"
In RL it is possible to use a wedge to putt by striking the ball just above center so that it does not lift off the ground but rolls along the green... in this virtual game however the wedge shot always lifts off the ground hence it cannot be described as a putting stroke.
In this particular case as it is a PUTTING competition I agree with the OP that only a PUTTER should be used.
Just my opinion but I would like to hear what the makers of this game think about the issue?
Rich