(I know this is naive, but) Corporations are made of people. All that money belongs to someone. So if those people decide to give that money to politicians, why does that make problems?
In general, publicly traded corporations are run by people aiming to get bonuses by maximizing shareholder value. Those shareholders could be very indirect, in particular for pension funds.
With this level of abstraction, the final shareholders usually are not (or don't want to be) aware of things done in their name. The situation makes it very easy to look the other way.
It makes problems because political hay can be made over how campaigns are financed. Most people want the gov't to have a large degree of control over how others conduct their affairs. Politicians believe that making reforms to campaign finance is a platform on which they can be elected. The issue is not whether campaign finance reform actually works, or improves the situation, the issue is whether you can be elected for promoting it.
People don't like the gov't but believe that it's a matter of who is in charge rather than the structure of the system.
Corporations are not "made of people". Legally, corporations have individual free speech rights like people (in the U.S.). People individually deciding to give money to politicians is different from the corporation giving money to politicians, but both acts are considered free speech in the U.S.
While you're logically correct based on that assumption, what happens when a corporation breaks a law? It's one thing if an individual within the corporation, but what a violation occurs due to a collective action taken by the company? Who goes to jail? Who exactly pays the fines owed?
With this level of abstraction, the final shareholders usually are not (or don't want to be) aware of things done in their name. The situation makes it very easy to look the other way.
IMHO, Adam Smith would be disgusted about this.