Thanks for responding. This is a problem with the applications, not with the map data. This situation is documented on the wiki here: osm.wiki/Good_practice#Don't_map_for_the_renderer

You're right that many applications don't distinguish paths with the access=no tag. Here is an example of one such path: way/503291111 (more: https://overpass-turbo.eu/s/1jNP). openstreetmap.org and OsmAnd distinguish private paths for some but not all map styles. Komoot does not distinguish private paths. I will file bugs against those projects/map styles.

I will not re-add it, but this path should not have been deleted. It physically exists. The access=no tag can be used to indicate that access to a trail is prohibited.