Earlier quoted context omitted.
Presumably Rand Paul, like many Republicans (and no small number of Democrats), accepts the idea that people can exercise their rights through government-registered entities which they own, called "corporations". These are entities without intrinsic rights, but a well-known intellectual framework called "corporate personhood" which dates back to some of our earliest jurisprudence in this nation says that you treat th…
While not debating the facts of your statement, I would contend that the only legitimate exercise of an owner's rights in the context of a corporation are to seek profit through commerce. Political speech, religious beliefs, and other "personhood" concepts are already granted to corporate owners as individuals.
Not all extant corporations are made to seek profits. There are nonprofits out there for a variety of charitable causes, social-justice causes, and other activities which often have a nontrivial component to them. I believe the ACLU is a corporation, for instance -- the EFF as well, and Amnesty International. We've seen many of their free-speech activities on this site. What provision shall we make for these activities and organizations in an alternative legal framework?
More generally: if an arbitrary profit-seeking corporation doesn't have first-amendment rights, why do newspapers specifically have them? (Or are we abandoning freedom of the press as well? That'll take some Constitutional doing, naturally.) Assuming that it remains, why should Rupert Murdoch's News Corporation or Bloomberg's Bloomberg have rights that Warren Buffet's Berkshire Hathaway doesn't? And where and how should we go about drawing a line between "press" and "not press" organizations in a complicated world where, for instance, an Internet-technology firm like AOL has bought a major news entity, Time Warner?
Outside speech questions specifically, if a corporation does not have first-amendment protection to speak as it chooses, does this mean the government ought to be legally compel its choice of speech? (e.g. endorsing specific politicians, policies, etc.) Is it protected from search and seizures by the Fourth Amendment? (Or, hell, from quartering troops by the third?)
These questions may or may not have good answers. Either way, these narrow margins can hardly contain a proper treatment of the matter. I'm just throwing a few monkey wrenches into the problem, as your comment was simple and straightforward, but the consequences of the proposed legal regime are complex and wide-reaching. :P