A path in OSM is not first and foremost a "walking path" - it is just a path that exist, and who is allowed to do what on that path is something we have other tags for. Same for private driveways. We have a right to record this information, and interfering with our right is vandalism.

We are very interested in depicting the reality on the ground, and we do not want to mislead people into believing they could walk somewhere where they can't. I know for a fact that AllTrails will respect the access=private in routing, i.e. it won't suggest you walk there; as for their cartographic representation, I don't know how they process access=private but if they really don't do that then it is a mistake on their part! For what it's worth, AllTrails themselves recommend using access=private: https://support.alltrails.com/hc/en-us/articles/360058130612-What-if-a-trail-is-on-my-private-property