Cornell cowboy, your explanation looks right to me. One can certainly have a private road over public land. However, in my experience you probably cannot prevent people from walking on the road or perhaps biking, and almost certainly not crossing it. I am most familiar with state trust lands in MT where folks have purchased motorized easements across trust lands to access private parcels. In addition, state land lessees can drive across state land where the public may not; I think this is a a specific corollary access granted in the lease but I’m not sure.
There are many, many examples of roads on federal land where certain designees are allowed exclusive vehicular access. The question that we may be getting hung up on is what rights do the public retain? In your case perhaps the easement (and road) are exclusive but in most cases I’ve seen the public retains whatever rights they had to begin with in the underlying land, which would include non-motorized travel. Is that indeed the case?