I don't remember the paths down there explicitly by now anymore. Are they really seriously overgrown to a mostly unusable state? For the legal state, the access tag is used, otherwise OSM documents what's on the ground. Abandoned is used for "features that have fallen into serious disrepair and which could only be put back into operation with expensive effort".
Generally, the same argumentation applies as I made in Sihlwald, the difference is that I couldn't verify the state of many paths there (except those i tagged with foot=yes) and therefore didn't change the tag.

I think the barriers should also be rendered explicitly but at least Carto fails to for the bar barrier. Gate would be be rendered but wouldn't be appropriate here as it's not meant to be passed by anyone.
However, having the paths on the map can also be meaningful, for example: It can help for navigation. And it can serve as information and help in route planning for people who have walked there before when they were still accessible and know they exist.