Are you sure that you are not-conflating private ownership with a legal access prohibition for pedestrians here?

It is very unlikely that Strand Lane is "cars only" as there is no such restriction in the UK. It was already tagged motor_vehicle=private by another mapper.

As I have not been to Strand Lane since I was a student at KCL 30 years ago, I have added a note for someone to check this - https://www.openstreetmap.org/note/4643776