A corporation is a type of association and has rights that are derivative of its member's rights. An association is composed of people. The only reason why a corporation (or other association) has First Amendment rights is because it is exercising the First Amendment rights of its owners and members. The personhood of a corporation is a legal fiction. The personhood of the ultimate members or shareholders of a corporation is not.
Corporations rights don't trump the rights of people. They are subject to the people for a reason, they are organizations that exist for the people, of the people and by the people. This is why we ALLOW them to exist with charters to operate and the discretion of the people. And by the way that part has also been corrupted, charters essentially don't even matter now.
But people have rights and just because you organise together as a union or a company, the government can't use that as a backdoor to limit your rights (Citizens United).
Why not? If you organize together that means you have more power than an individual. Governments can and should certainly try to equalize that imbalance where possible and where it makes sense.
It was a normative statement I believe ("companies should not be people and should not have rights") rather than a factual one (I'm sure they're aware & that's why they felt compelled to make the declaration). English is sadly lacking in this regard, where you often get to make a strongly worded statement or an unambiguous statement, but it's very difficult to do both at the same time.
The funny thing is that corporations have a bunch of advantages like limited liability on top of the US Supreme court ruling them to have the same rights as people, which means that corporations actually have more rights than regular people.
I mean, doesn't that necessarily follow from an implicit understanding that corporations are ruthless profit maximizers though? I fail to see the supposed ideological inconsistency...