could give essentially unlimited money to campaigns
Wrong. It is illegal for corporations to give money to politicians or campaigns, in any amount. You can't even provide discounts to those politicians (that's a "contribution in kind").
When people talk about corporate money in campaigns, they're either:
(1) Misreading donations by employees as donations by the corporation (you are required to report your employer to the FEC when you donate to a campaign, and this is public information)
(2) Permitting corporations to publish ads with political content, which indeed, the First Amendment prohibits Congress from restricting this (whether corporations are people or not does not change this, 1A says "no law", it's not a right bestowed on people only).
Yes, if you bothered to read the US Code, you wouldn't have to get very far to understand this, as this is provided in the LITERAL FIRST PARAGRAPH. Not only is this not a new phenomenon, it's a foundation of Common Law too. In the law, corporations and persons are synonyms so that the law applies equally to both of them. You want corporations to be accountable to the law, right? That's what corporate personhood is for.
suddenly Republicans want to kill the messenger
No, corporations have the right to say whatever they want. That doesn't change the fact that if corporations want to enter into politics and try to sway the discourse, then by all means, let's play politics. Play stupid games, win stupid prizes.