Lots of important precedents rely on the First Amendment as applied to corporations
Many opponents of the decision argue that the First Amendment provides no protection whatsoever to corporations, but is instead a protection of an individual right: the right of individuals (and unincorporated groups of individuals) to speak and write without government censorship. Since corporations are not people, this guarantee does not apply. Thus, the argument goes, the government may regulate their activities, including their publications.
But if this principle were adopted, a lot of important free-speech precedents would be in jeopardy:
New York Times Co. v. Sullivan (1964) held that the First Amendment protects the right of the New York Times Company, a for-profit corporation, to publish material critical of public figures, without fearing libel sanctions unless its statements were both false and made with "actual malice".
New York Times Co. v. United States (1971) held that the First Amendment gave the New York Times Company, a for-profit corporation, a right to publish the leaked Pentagon Papers.
Hustler Magazine v. Falwell (1988) held that the First Amendment protected the right of Hustler magazine, a for-profit corporation, to publish a satire ridiculing Jerry Falwell without being liable for "emotional distress" damages.
There are many more important precedents, involving commercial book publishers, magazine publishers, film-production and -distribution companies, and more. Unless all these precedents are to be overturned, or some alternate way of regrounding them is proposed, I can't see the argument that the First Amendment does not apply to corporations at all as a particularly serious or desirable one.
Do we really want a result holding that the government may engage in unlimited censorship of anything published by a corporation, including, for example, book publishers, newspaper and magazine publishers, film distributors, etc.? That would be a stunning reversal of the past century of free-speech jurisprudence.
Interestingly, corporate personhood has actually not much figured in any of these precedents. The First Amendment by its text doesn't protect a right of people, but is phrased in the negative, prohibiting the government from enacting certain kinds of legislation: "Congress shall make no law ... abridging the freedom of speech, or of the press". Civil libertarians have typically argued that this ought to be interpreted as, essentially: "Congress may not censor, ban, or otherwise regulate speech or writing", subject to very narrow exceptions such as the fire-in-theater example. As Hugo Black famously put it, "no law means no law!"
So, where's that leave us?
I nonetheless agree that there are significant problems with the way campaigning for elections goes on in the United States. But I don't think it's an issue of corporate personhood or can be resolved by simply saying that the First Amendment doesn't apply to the activities of corporations.
Rather, there would have to be a way to carve out a narrow exception for specific kinds of electioneering. That's essentially what the now-overturned Austin decision (1990) attempted to do, but I think generally unsuccessfully, as evidenced by the fact that neither side in the present case attempted to defend Austin's reasoning: one side argued to overturn it (they won), while the other side conceded that its reasoning was suspect, but argued instead to reaffirm its central holding via a new rationale (two alternate rationales were advanced).
Such rationales might well be plausible. Perhaps this particular issue constitutes a narrow exception to the First Amendment: a plausible holding could have been that direct electioneering by corporate entities in the period immediately preceding elections is an exception to the general prohibition on government regulation of publications.
Another occasional proposal is that "the press" refers to a particular kind of corporation, like the New York Times Corporation, that should enjoy First-Amendment protections that other corporations do not. This would save the NYT precedents, and possibly the Hustler one, though "the press" would have to be interpreted pretty broadly if, for example, film-production companies (especially those producing films other than documentaries) are going to somehow count as "the press" too. It would also produce something of a demarcation problem, and I'd be wary of who will be deciding which companies count as press and which don't, given the large gray areas. But it's not an absurd distinction on its face, even if it's one civil libertarians would rightly be wary of.
The current discussion, though, seems to mostly focus on a "Corporations aren't people!" argument, which I think is a bit of a red herring. Whether or not corporations are people, the First Amendment does not permit unlimited government regulation of publications produced by them, and it would be quite undesirable from a civil-liberties perspective if it did: at least, that is the case unless someone has a good argument for why civil libertarians should want the above cases, and others like them, to be overturned.
In short: It may be that the majority opinion in this case went too far, but the opposite proposition, that the First Amendment is no obstacle at all to any kind of government regulation of publications produced by corporations, is even worse.