Some people have argued that (1) statutes and constitutions should generally not be read in ways that render particular provisions superfluous, and (2) therefore the Free Press Clause should be read as providing special protection for the institutional press, beyond what the Free Speech Clause provides for other speakers.

I generally agree with point (1), but I don't think that point (2) follows. Protection for the "freedom of speech, or of the press" can quite sensibly be understood as ensuring that both speech (spoken words) and press (printed words) are to be equally protected (and perhaps other communication would be protected as well).

Without the Free Press Clause, the First Amendment might have been understood as not covering material that is printed and thus capable of being broadly disseminated. (One can imagine a government official arguing that speech, or even a handwritten letter, is all well and good, but printed material is much more dangerous; in fact, English history had been full of similarly justified restrictions on printing.) Without the Free Speech Clause, the First Amendment might have been understood as only covering material that is printed and thus capable of being broadly disseminated. Reading both clauses as protecting the same context, albeit in both media, doesn't make either provision superfluous.

Indeed, modern discussions of freedom of speech often cast it broadly enough to cover all communication, whatever the medium. But this broad understanding of the provision was likely itself molded by the breadth of the "freedom of speech, or of the press" language. To the extent that even newspaper publication is often described as protected under the Free Speech Clause, that's so precisely because the accompanying Free Press Clause has created a legally culture in which printed speech is as seen as no less protected than other speech.

I should note that, even independently of the above, I don't think the "press" must refer to the press as a business, as opposed to the press as a technology. But in any event, "freedom of speech, or of the press" strikes me as providing equal protection under both clauses, not special protection under one or the other.