Andy Gelman is usually right about most things, but here, he manages to be both right and seriously, nay, fundamentally wrong.
The problem with betting markets is the same problem with the price system in all (asset) markets identified by Stiglitz and Grossman in 1970s: the price system conveys information cheaply, while real expertise is expensive. If the information conveyed by the markets becomes too perfect, people simply underinvest in the real information. And the underinvestment continues until the market information is just wrong enough to justify investing again. This is further compounded by the fact that, in the end, polls are themselves expensive and unreliable as conveyors of information. So reliance on the markets, imperfect as it might be, is bound to weigh in even than asset prices in financial markets. An echo chamber is likely to form and to sustain itself. Unpredictability in politics is inevitable and no amount of data can eliminate it wholly. By trusting too much in markets or polls, we are setting ourselves up for mistakes.