Thanks for your reply Alejandro.
I didn’t realise I could use the area tag with this sort of thing.
I’ve marked one area with the following tags:
access=private
area=yes
highway=service
surface=asphalt
…and OSM seems to be fine with that. It now shows on the map as a large white area which is part of the service road it leads off.
So now I’ll look for the other similar areas and continue in that fashion.
Thanks again.