I was very wrong, because Caplan spends a few pages near the start of the book utterly demolishing this argument as a sufficient explanation for why democracies sometimes do stupid things. The short version is that if the ignorant majority votes randomly they will tend to cancel each other out and if there is a minority which votes rationally they will end up dominating the outcome, since after all Aumann's Agreement Theorem says that rational people should always agree with each other. Ok, but more seriously you would still expect that in areas where there was fairly strong expert agreement (global warming is man made, minimum wage laws hurt the worst off, etc) we would see large number effects give victory to the experts.
So it requires systematic bias for the public to disagree with the experts and Caplan spends most of his book talking about biases and why in economics we should believe the experts instead of the lay public. He also mentions toxicology and how people tend to want to treat danger as a matter of contagion rather than "the dose makes the poison". I'm sure I could come up with more examples from other areas, but I'm also sure that you, gentle reader, can come up with your own and even if we can't cite statistics we might as well go on.
In the last part the book discusses some ways that this effects politics. Politicians want to be elected, and its easier to do that by confirming the biases of the voters rather than arguing with them. But once elected they want to be reelected, which means that they have to ensure that things go well, which means that once elected they have to talk and listen to experts - and then maybe reverse themselves on things they promised to get elected.
So the upshot of the book seems to be that lying politicians are better than the alternative, who actually carry out all the promises they made to get elected. Not a pleasant conclusion and one I'll probably blog more about later. But despite my job of summarizing it here you should all still read the book.
(1) At least Megan McArdle and someone else had said good things about it at some point, if I remember correctly.