Eric, It generally rests on: the quality of the course, smart, short transitions in the routing, and it's ease as a walking track. If the experience playing it is challenging and memorable, given the other elements are in order, I think you certainly could have a "Walking Preferred" facility. I would always hesitate to dictate the "only" part. There are folks who, for a vareity of health/age reasons, really couldn't enjoy their golf without the assistance of a powered cart ...why would we want to deprive them of the chance to play your great track?!
Mike makes a good call on the no path idea. A possible compromise, to minimize excessive turf wear and poor asthetics around high traffic areas, might be the use of naturall products around transition /tee areas where limited cart play could be permitted. I've seen: crushed coquina, smaller, fine-mixed aggregate, larger base-rock material covered with medium sized stone and double ground wood chips, all used with good results as a path surface, depending on soil/drainage conditions. The key is to adapt materials to what the course/weather conditions will allow and keep the look consistent and compatable with the terrain.
Your dream is certainly doable! Keep investigating, compiling information and talking to folks...you might be surprised how quickly some financial support could develop if the ground is as good as you believe it is.
Cheers