The landowners have expressed legitimate safety concerns with climbing in the gorge. The site contains a popular public road and the potential for passers-by to be hit by falling/dislodged rock is obvious. If you wish to climb or scramble in the pass, BMC advice is to limit visits to either very early or late in the day, midweek.
Open access land, designated under the Countryside & Rights of Way Act (2000) give area access rather than linear access as provided by public rights of way. It also gives a legal right of access specifically for climbing, as well as walking and other quiet recreation on foot.
Please bear in mind however that the landowner still has the right to restrict access for up to 28 days per year (often used on public safety grounds for shooting in moorland areas), and can also apply for longer term restrictions with Natural England (such as bans on dogs, or regular restrictions during particular times of year). It is important to check for these restrictions regularly as they can be added at short notice – all details for open access land in England can be found on Natural England’s website.