Hi, Welcome to OpenStreetMap.

It is likely that the first mapper of the path was aware that bicycles were not permitted. Also the landowner might not want cycling on their land. Therefore, unless you to definitely know that bicycles are allowed then the tag should remain. Routers don't always work as desired so we should not alter OSM data to try and help them. A good router will not route a bicycle through a bicycle=no tag.

Regards Bernard.