Simply put, there are roads that are legally designated public roads that cross private land to get to public land beyond,...and then there are roads that look just the same on a map that are not legal public roads. The private landowner through which those roads cross unfortunately has every legal right to close those roads to the public.
On federal lands, the administering federal agency should be able to tell you whether a specific road is a legal, public road. Same with state lands. Just because a road that goes onto private land goes out the other side onto public land again in no way indicates whether that road is legal for the public to drive (or walk) or not. Some are legal,...some are not. It all boils down to whether a legal, documented easement for public access is on file or not.
This has become a major problem in the western states. Many roads that have traditionally been used by the public to access public lands are being closed by landowners because there is no legal easement for the road on file. More and more, greedy, unscrupulous people are buying up properties with the specific intent of closing historic access routes to public lands to, in essence, create private playgrounds for themselves on public property.
Unfortunately, in many cases, the public land management agency that is supposed to be looking out for the public interest is turning a blind eye to this practice. Bottom line is there are tens of millions of acres of public lands that are owned by you and me that have essentially been privatized and to which you and I have absolutely no access.