I can't think of anyone that has made it a point to want to travel that short section of road, unless they wanted to visit the two families living up there, and that is rare. It's a dead end road, and the only thing to see are some old tailings from a mine that petered out decades ago. I also can't think of anyone that would be out anything were the road abandoned, and the gate left standing. If you've ever been up that way, you aren't missing out on much.
As far as the road "providing legal access to public land and waters that could be used for public recreational use", that "public land" is a narrow road walled-in by a narrow canyon, and the "water" is a creek about a foot wide and about two inches deep. Like I said, I can't see very many people missing out on much by not being able to go up there.