Calling a corporation "people acting together", or just "a group of people" downplays the fact that corporations are indeed made up of many people but with only a very few of those people able to exert any real control over the whole entity.
Most large assemblies of people are like that. Unions are like that. Political action groups are like that. Underground movements are like that. The impracticalities of having everyone agree on everything means that whenever you have a large assembly of people you must hand over control to decision-makers.
Shareholders do have ultimate control over corporations, by the way. But shareholders, generally, would rather sell their stock--or fall asleep at the wheel--then exercise corporate governance.
But that's besides the point.
You are downplaying the fact that the Constitution clearly grants us right to assembly, in the same amendment that grants us free speech, and an assembly of people has the right to speech just as much as a person does. According to jurisprudence, the government may only curtail these rights with laws that meet a "strict scrutiny" standard: they must meet a compelling government interest, they must be narrowly tailored, and they must be close to minimally restrictive. McCain-Feingold does not meet the second and third standards, and 4 justices think it does not meet the first.
But, with a political coalition, people join the coalition in order to promote the mission of the coalition. With a corporation, people join the corporation in order to get paid, not in order to promote the viewpoints of the corporation. When you conflate the two objectives of profit and social decision-making, you're implicitly encouraging people to sell out their values. That's not democratic.
You are also downplaying the difference between a group of people and a corporation. A corporation is when group of people place some of their rights and responsibilities into an artificial entity, in order to promote its own interests and shield the creators from a certain amount of responsibility. Why must this political-economic golem be given the right to free speech? My contention is that the right to free speech of the creators and shareholders is completely sufficient. Giving the controlling shareholders access to non-controlling capital to promote their own viewpoints is troublesome because it causes several interests to conflict that don't need to.
When you conflate the two objectives of profit and social decision-making, you're implicitly encouraging people to sell out their values. That's not democratic.
This is also a problem when the government takes money from some and gives to others. A socially conservative union member may vote democratic because he wishes for the government to restrict the rights of his employer and transfer wealth from customers to him. A social liberal working for a defense contract might vote republican because he wishes for the government to transfer wealth to his employer (and indirectly to him).
Anytime the government transfers wealth from some to others, it implicitly encourages people to sell out their values. I don't see why it is more of a problem for shareholders than for union members, people on social security/welfare/unemployment, or employees of corporations.
"Why must this political-economic golem be given the right to free speech?"
Why must this political-economic golem be subject to specific laws, regulations, and taxes beyond the ones all the people in a corporation are already subject to?
The answer to your question is because it's subject to special government attention, the people involved with such a device have a right to use it in the political process to protect their interests and involvement with it.
As for selling out, people make decisions all the time. If you don't like the politics of your company, try to do something about it or find another you like better. (Realistically, this isn't much of a concern. Corporate donations tend to be remarkably even-handed so as to not give the winner of an election the impression they preferred the other guy.)
Think about it, folks. A corporation is a tool, like a car. There are laws that apply specifically to operating cars - you can get ticketed or arrested for breaking them. However, laws that said, "You can't use your car to drive to court in order to contest a traffic ticket or to drive to voting locations to vote for fewer no-parking zones" would obviously be abusive.
