There seems to be a resounding consensus on the internet that private campaign funding is a bad idea because it usually leads to corruption and unfairness. Are there any good reasons to think otherwise given that in the status quo election campaign funding is allowed?
There are two economic theories/explanations for advertising. The first is that advertising is informative; it conveys relevant information to people who would otherwise not have it. This makes the market more competitive and leads to more people getting what they want.
The second is that advertising is persuasive (a discussion starts on page 3). In this case advertising actually effects people's preferences. If advertising is persuasive, then companies are stuck in a game where they must spend money advertising to persuade, but their competitors do so as well. In the end, this leads to an equilibrium where every firm spends money on advertisements and the consumers have to pay for them in higher prices.
The same holds for political advertising. If the effect of the ads paid for by the massive private campaign contributions are more persuasive, then your 'resounding consensus' is correct. If the ads are more informative, then your resounding consensus is incorrect, and limiting campaign contributions will just lead to fewer voters getting the information they need.
Personally, I find political ads to be almost without exception both unpersuasive and uninformative, so my intuition doesn't seem to be helpful. The empirical work in this matter is tough, since quality candidates also attract money; a tricky endogeneity problem. One attempt to find the relationship between campaign spending and election results is Steven Levitt's 1994 paper which uses cases where the same candidates face each other in US House elections to help get around the issue of comparing apples to oranges. His somewhat surprising result is that money is fairly ineffective at swaying elections. If this is actually insightful into how voters make decisions, then there's little risk in private campaign contributions swaying elections. There's some inherent logic to this as well; if voters were actually stupid enough to be swayed by black and white photos with silly character assassination or contentless dribble with patriotic music and bald eagles in the background, then democracy has much deeper problems.
An important corollary to this is that politicians would have to be smart, notice that adds were ineffective and thus not pay favors to their campaign contributors. This might actually be the trickier sell, but corruption is already somewhat tough to pull off unless it involves billions in farm subsidies.
Finally the risks of having public campaign finance is that we might end up with the government stifling rising stars; if someone's not big enough to get public funding and not allowed to use private funding; they might be stuck on the outside never able to get their good ideas into the race. And the flip side to the coin, with public funding there's the risk of large amounts of public money being wasted on candidates with no chance of being elected and nothing to add to the debate.
Edited: I forgot one major point. Independent of the other arguments, many (including the US Supreme Court) consider the ability to buy TV spots, signs, and bumper stickers a fundamental part of free speech. The philosophical position being that free speech is so essential to a free society that giving the government power to regulate it endangers the very core of a democracy. Limiting private campaigning is tricky; who's to say what's campaigning? Could the fact that politics stack overflow hosts a question asking what experience Donald Trump has in government count as materially contributing to Hillary Clinton's campaign? Those who support this argument would rather not risk tyranny in the many regulations accompanying regulation of campaign contributions. A more moderate position along these lines would hold that campaign finance laws should be as simple as possible and err on the side of free speech.
All that being said, I don't think any of these arguments are airtight and reasonable people can disagree on this issue.