If there's a physical path there, but people aren't supposed to go there, marking it as private rather than deleting it allows for people using maps based on OSM data to see that they shouldn't go there, rather than maybe confusing it for some other path or thinking that it was a path that just hasn't been mapped yet. It can also help with orienting oneself ("I walk past the closed path, and take the next left" and such). You may want to take a look at these wiki pages:

https://wiki.openstreetmap.org/wiki/Why_can%27t_I_delete_this_trail%3F

https://wiki.openstreetmap.org/wiki/United_States/Trails_Stewardship_Initiative