From https://www.parks.ca.gov/?page_id=29709 (California State Parks HCRSP website):
"Bicycles are allowed only on Pipeline Road, Rincon Fire Road, Ridge Fire Road, and Powder Mill Fire Road."

From https://wiki.openstreetmap.org/wiki/Key:access (OSM's own wiki about the access tag):
"Access tags pre-eminently describe LEGAL permissions/restrictions ... regulation, rather than guesswork. They do not describe common or typical use, even if the signage is generally ignored."

As these trails are not explicitly granted access, and the same at UCSC (I have worked personally with their Transportation And Parking Services on properly mapping campus access tags in OSM, and can similarly document their explicit mountain biking paths, now properly denoted in OSM), the access=no tagging as it presently exists appears to be correct.