I think you've identified the problem quite well. I don't think the carte blanche "public financing" is a sufficient solution however. If you look at countries with public financing, it's not as if money is any less of a problem in their elections. The biggest wallet is still the strongest competitor. Even without PACs and SuperPACs buying up the airwaves, even if everybody knew everything via the most democratic form of communication, i.e. Internet, there are still numerous ways for money to enter the election (in the latter case, via astroturfing).
Running for office is currently a popularity contest. In fact, the very mechanism is called "popular vote". The best person doesn't necessarily win. Rather, the most popular person does. Popularity comes about in multiple ways, but in the end, it boils down to marketing. Marketing is not necessarily telling the best-sounding lies (though it's likely the case considering these are politicians we're talking about). Marketing also involves raising awareness and manipulating the narrative. Any campaign is dead on arrival without a good marketer with a good marketing strategy, knowing who to say what to when.
A good marketer requires money. Or promises of benefits. We're all intelligent people here. We all are talented. We all command a price (though money is but one type of payment), and understand and implicitly acknowledge that the price of our talent is higher than the price of those with inferior talent, but also lower than those with superior talent. There's no reason to believe that this does not apply to marketers.
The only way to remove money from politics is to remove popularity from the process. There are many ways to blunt the impact of money (public financing being one such, spreading money around more evenly is another), but so long as there is value in talent, money and power will remain correlated.
Now, as for the methods of reducing money's influence in a popular election, those would be public financing, reducing income disparity (the two I previously mentioned), improving education, and democratizing communication. Tying the number of representatives to a fixed population size (rather than fixing the number of representatives and floating the population represented) will also eliminate other corrupt practices like gerrymandering. Going to a ranked voting system would also help, but that's more to eliminate the two-party dominance. These last two reforms would indirectly reduce the amount spent per party though they would not reduce the total spent nor the impact per amount.