In the US politics has very much become about money- how much is contributed to a campaign's finances is a major issue because of the cost of advertising. In the EU, however, it is common to ban political advertising to stop money becoming more important than the ideas.
My question is, why is this form of advertising allowed? Have there been attempts to ban it? This question is related specifically to the US, but examples of other countries would be a bonus.
The key federal (Supreme Court) case law here is Buckley v. Valeo, decided in 1976. In 1974, President Ford signed a campaign finance law to do exactly what you suggest - limit the amount of money that could be spent on political campaigns.
The Supreme Court, however, decided that the law directly conflicted with the First Amendment protection and right to free speech. They wrote:
"The Act's contribution and expenditure limitations operate in an area of the most fundamental First Amendment activities. Discussion of public issues and debate on the qualifications of candidates are integral to the operation of the system of government established by our Constitution. The First Amendment affords the broadest protection to such political expression in order 'to assure (the) unfettered interchange of ideas for the bringing about of political and social changes desired by the people.'”
In the same way that corporations can be people :) money is speech:
"A restriction on the amount of money a person or group can spend on political communication during a campaign necessarily reduces the quantity of expression by restricting the number of issues discussed, the depth of their exploration, and the size of the audience reached."
Because SCOTUS has said you cannot ban campaign spending as a free speech violation, the solution is to federally finance campaigns instead. In theory, this mitigates the effects of money, and even can encourage realistic candidates who otherwise do not have the funds to run.
Typically, major elections are now fund-limited by agreement. The United States Federal Electoral Commission provides significant funding (derived from those checkboxes on your income tax return 'Do you want to donate $3 to presidential election campaigns?') if and only if:
You are a "major" candidate, meaning you or your party has, in a previous election, received votes over a particular threshold.
You receive a certain amount of funding from donations (this is a matching program)
You voluntarily agree to certain restrictions on the total amount of money you will spend.
These three pillars of the campaign finance system essentially ensure that the fund isn't depleted by marginal candidates. That last point is key - campaigning financing is a bribe incentive to voluntarily adhere to limits to reduce the monies involved from all sources.
@Affable Greek gave a practical legal answer.
For a more idea-based reasons, there are 3 approaches of seeing the necessity of political campaign ads and the need to spend money in campaigns:
Incumbent politicians have a basically insurmountable way of campaign advertising via being an incumbent.
They have the name recognition. They can do favors to electorate with taxpayer money spending, or via passing laws/rules/regulations favoring targeted electorate parts. They get to go in the media and speak out, as an active politician.
ALL of this is denied to a challenger, whose only way to overcoming this advantage of an incumbent is political advertising.
Similarly, in a contest between two candidates, if there's no campaign advertising, the one with most current celebrity status and recognition wins. President Kardashian anyone?
More generically, money is basically just a method to be more persuasive by increasing either the volume (eyeballs) or quality (fancy targeted ads based on detailed research) of someone's political speech.
Sure, money makes things "unfair" because the person who has more money can have more effective political influence on the populace.
However, if you want to make things 100% fair, then you need to remove all OTHER things that make political influence unfairly more or less influential - for example
a silver-tongued demagogue extroverted lawyer with advanced rhetoric talent has an enormously unfair advantage (in both persuasiveness and quantity of his political influence) compared to shy, tongue-tied, loner numbers geek with no great speaking/presentation/rhetoric skills.
Someone who doesn't work, has 16 hours/day to propagate their political views. Someone who does, only less than 7 (I don't have cites for it but surely that is the reason that since 1960s the college students have been such an influential force in politics).
Someone who works at mass media can spread their political opinions at 1,000,000x the volume of a single person (for example, witness all the media's attempts to create a positive image of Hillary Clinton on TV - such as the latest TV series about heroic female Secretary of State). Or NBC "News" non-stop pro-Obamacare coverage when it was being debated. Or gazillion other examples of media providing biased information to suit their political views and goals.
So what you're asking, if your goal is to get money out of political advertising, is to prohibit just ONE way of how political speech is "unfairly" magnified and leave all the others.
The political system has an instability to move toward unlimited campaign funding. This is particularly a problem in the US, because the legal system there is far more under the control of electorate than it is in most other Western countries. This means that you can start from a situation where the law does have some restrictions on campaign funding but then people get get elected who have used all that the law allows and they can then use their influence on the legal system to expand the rules, leading to more such politicians to get elected and the rules get even less restrictive on campaign funding.
In most other countries there are very strict laws that prevent this from happening. There is no way you can invoke the right to free speech to argue for expanding the rules. What is also restricted is what you can say during a campaign. If you tell lies and win an election, your election victory can be overturned in court, like in this case.
