As it is now, marked only as a route, it is no actually shown on most map styles. Possibly only on dedicated snowmobile route map. In my opinion, the routes could marked with highway=track (meaning, the surface is not treated for driving over, I assume that's the case) with addition of marking access=no to ban all vehicles generally, and add snowmobile=designated. (see https://wiki.openstreetmap.org/wiki/Tag:access%3Ddesignated) 
That way, the routes would be shown or common map styles.