"You can not copy the map. But you can take all the street names and locations off of it and draw your own map."
Correct. But if you're going to do that, you might as well go back to the aerial photos and/or GPS traces and/or public domain data which the OSM map was created from.
To successfully extract all the public domain information out of OSM, you'd almost have to do as much work as just starting from scratch.
It's similar to the fact that you could, in theory, extract all the raw facts out of Wikipedia and then use them to recreate your own encyclopedia, which wouldn't be subject to the copyleft requirements.