Japan Rail is public-private and many of the other train lines are fully private. "Public" is kind of an empty distinction here, Americans associate the two concepts because they think mass transit is a kind of gift you give to poor people instead of something everyone actually uses.
But there is plenty of need for car-shaped transit in Japan and people take taxis and use cars all the time. You might have luggage/equipment to take somewhere, it might be raining and you don't want to walk the last mile, etc.
(It's surprisingly hard to take luggage through transit in Tokyo. For instance, maps apps won't give you a transit route that uses elevators, even though everyone with a baby carrier would use it.)
