If I remember that path correctly, it starts in the parking lot near the bag drop and allows golf carts in the cart parking area to access the starter shacks for Caroline and Virginia nines. Amelia can be accessed through the parking lot. So it appears to be very much part of the golf course operation. Since it was apparently a starter that told the OP not to use that path to walk on, I suggest going into the pro shop and have a conversation with the head golf professional.
I would have no idea if walking is permitted there. Good luck getting a satisfactory answer.
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