It is a different case when mountains are privately owned. While I would like to climb every mountain that is within the range of my climbing ability, I understand that some mountains lie on private property. While I would like to get permission to climb these mountains, I understand that some landowners don't want people on their property. That is their right, because private property rights are one of the foundations of our free society. I respect private property rights. Many of us own our own homes, and I don't think any of the readers of this forum would want strangers to walk around without permission in their backyards or in their houses.
For many people, particularly people living in rural areas (where most mountains are located), their land is what they rely upon to earn their living. Therefore, they have a vested interest to ensure that people do not enter their land without permission.
In the case of the Navajo sacred peaks, I have climbed Humphreys and La Plata, and they are located on National Forest land, which is public land, owned by all Citizens of the USA, set aside for recreational purposes. Therefore, anyone can climb them. I am not sure of the status of the other two peaks. If they are located on Indian Reservations, then the Indians own the land, and they have the right to restrict access.