You will be liable for any damage even on a private road.
Having lived on an unadopted street, which is technically similar to but not the same as a private road, I would go with mixing ballast and cement. You’d be consolidating the surface not filling the hole with fresh material.
Our road was a mass of potholes, blocking one hole changed the water flow and flooded other areas.
Only when the residents all got together did it get resolved. They bought several truckloads of stone to resurface the whole rd.