Please see the definition for designated (https://wiki.openstreetmap.org/wiki/Tag:access%3Ddesignated). It suggests that while a motorist may not be permitted on the pathway, a different user (pedestrian, cyclist, skateboarder, etc.) may use the path. However, the designated definition does not apply here to begin with given that a person cannot access the path due to a guard gate preventing any user on any form of transportation to get in without permission. Accordingly, the  section "Allowed Access" should not note foot or cyclist as designated, but as private for only individuals who are permitted into the area