Out of an abundance of caution, I asked two communities, the OSM Community Forum, and the OSM Reddit about how to correctly tag the pathways we were discussing. Members of both communities have stated that because the pathway is within a gated community, the pathway should be labeled as private, not designated because the designated tag overrides the private tag. It appears that Spaghetti Monster's comments above fall in line with those of the other members of the OSM community. You can read each community's messages regarding this at the below links and post your feedback there as well:

https://community.openstreetmap.org/t/how-should-i-tag-a-foot-cycle-path-within-a-gated-business-complex/127153

https://www.reddit.com/r/openstreetmap/comments/1j7iqxh/question_about_tag_on_a_cyclefoot_path_within_a/