There are a couple of clubs up here that also don't like play off the fairway during winter. One of them insists that you tee up every shot, the other gives you a small piece of range mat that you have to place your ball on and play your shot off that. Needless to say they don't get much business during the winter months, but to be fair, they are in pristine condition come spring.
That sounds completely retarded. A propper iron shot off a tee produces a divot anyways...
And as for carrying a peice of mat around, well... i give up.