Levitt does not have an axe to grind as far as I know and has theories guaranteed to annoy both the left and the right (eg, swimming pools are more dangerous than guns, or Roe v Wade resulted in decreased violent crime). He's not a partisan, but he does offer opinions on a lot of topics that he doesn't know very well.
Politicians certainly behave as if money matters. It gets their message to the voters. And in a close election, extra TV advertising makes a difference. At least I believe it does. You would have to look more closely at the data. I'm not sure Levitt has done that.
Money is connected with establishment support. Can you imagine a candidate who is the darling of the establishment and who has no campaign chest? On the other hand, sometimes self-made millionaires like Steve Forbes launch campaigns with virtually unlimited funds, but it does them no good, because they're still seen as marginal. In cases like this, Levitt has a point that money by itself does little good.
I think what often happens is that money is part of a self-reinforcing cycle. Name recognition, endorsements from establishment figures, campaign contributions, media coverage -- it's all part of the image of a serious candidate. For disinterested voters, candidates without those things are second-class citizens. They accept the choice between mainstream options. Imagine George Bush running for president as a dark horse with a small budget -- he wouldn't have had a chance.
Incumbent congressmen almost always get reelected. But surprisingly, one place where dark horses with less money often win is in US presidential primaries. It occurred to me that the reason for this is that the decisive votes are often cast in Iowa and New Hampshire. Those states are won by retail politics at the grass roots, by shaking hands and talking with small groups of people. Under those circumstances, a candidate's ideas and presence are more important than money.
And it's one reason I hope that Iowa and New Hampshire, or a couple of other small states, always have the earliest primaries.