I don't believe the we have to mow more then the edge at all. I know a little about the subject, living in MN my whole life and dealing with the town ship for the same reason. Many people just want it mowed for weed control. It has nothing to do with clearing snow from roads. The ditch is the first to fill full of snow, the first time you get wind and snow. No matter if mowed or not. You can't tell which is which when you drive by in the winter... most grass is below grade. Mowing the edge is fine. All the grass in the lower ditch does no harm to drifting at all. Your ditch will be level with snow and drift any way, fact. It would save tax dollars as well. Many farmers want them mowed to keep people from hunting the roadside as well. Sorry but snow has nothing to do with it. Just my two cents.
I don't mow mine and neither does the township, except the edge. I don't see any big drift ever, it is worse in certain areas weather mowed or not. A fence will cause more drifting then grass in a ditch. Many farmers also in cattle country do it for the bales.