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.