It's not enough to only look at elections. The topics that the media discusses, and therefore the options that people are aware of and the issues people base their vote on are decided by mostly privately owned and increasingly consolidated media companies. Nobody will know about candidates that are not approved by some part of the elite in this media landscape. Any opinions that go against the interests of the media owning elite will not see much coverage. Sure, maybe money during elections does not matter that much, but elections are the very last step of the process of picking leaders, and the preceding steps matter as well.

Also, if money did not matter during elections, I doubt we'd see so much spending on them. Studies are being funded by companies and the wealthy as well, so a study or two saying money doesn't matter is not definitive proof.