Note that almost all Japanese railways started out as public sector projects, and were handed off to private sector to operate. Urban rail operations are within the wheelhouse of private organizations - but I don't think actually building them out is. At very best it could only be done with an extremely tight partnership between public and private sectors. With self driving cars, the private sector is taking more or less a unilateral decision to invest in them, something which cannot be done with pretty much any mass transit project.