Hi Jonathan and welcome to OSM!

Take a look at https://wiki.openstreetmap.org/wiki/Why_we_won%27t_delete_roads_on_private_property for a good explanation on why private roads shouldn't be deleted.

To mark the fact that it's private, you should add the tag `access=private`. To specify that it's a driveway, you can add `service=driveway`. You can find further tags that might be relevant for this feature in the Wiki: https://wiki.openstreetmap.org/wiki/Tag:highway%3Dservice.

Hope that helps, happy mapping!