Hello welcome to OSM. The way you removed in this changeset seems valid. If it exist on the ground should not be removed, regardless if its private or public.

Please read for more info: 
osm.wiki/Why_we_won%27t_delete_roads_on_private_property

The correct way to handle this in OSM is to use the tagging system. access=private seems to apply here. Once this is done services that do use OSM data will reflect that and will no longer route anyone on that road. Also the road name field should not be used at a description. The tagging system till let users know if a feature is private.

I have restored the road, and tagged it appropriately for your reference.