Jeff, that's a trivial workaround for the richest - buy or start a newspaper. Co-opt an existing union or corporation. Given that so few of these types of organizations that you mention are actually apolitical and nonpartisan, it doesn't seem tough to manage. Fixing the problem would be worse: I do not like the idea of some bureaucrat, likely put into office by one party or the other, going through a list deciding whose political speech is allowable and whose is not.
JECE, here's a big problem with what you're saying, at least for me. I believe strongly in some issue, and I want to exercise my right to do so honestly, peacefully, and with the consensual aid of some newspaper or TV station. You are saying essentially that because you do not like the peaceful outcome of me exercising my right to free speech, it should be limited. Even if I didn't think that was unconstitutional, just as a practical matter we should err on the side of protecting rights. Also, if we public financed elections, you wouldn't mind your tax dollars supporting, say, a white supremacist or Stalinist or some other terrible person?
Think about what you're all saying. People want to organize and make their voices heard in politics, and they want to do without risk of retribution. That is the essence of free speech. Saying you have this much free speech, but no more, is wrong.