But the NYT itself still makes legal decisions, like which articles to run. Lets say they decide to run a particular article which is later claimed by the government to be libelous. Since the NYT has no rights under your ideal the government could simply find them guilty of a felony without giving them due process of law and end their corporate existence.

I would argue that so long as corporations continue to exist they require a minimum amount of rights, such as a right to due process, or else they just become tools for the state to act without any of the restraints our current system has developed.