The media spent a lot of time discussing Palin's intelligence presumably because many voters cared about it.
Not because the voters cared about her intelligence per se, but because demonstrating that she's a moron constitutes an argument the Republicans can't easily defend against. It was pretty obvious that Palin isn't exceptionally intelligent - far more obvious for a typical voter than any DNA test could be - and still the debates weren't any less irrational than usually.
I would also expect soon emergence of social defense mechanisms against such tests. Saying that somebody is probably an idiot based on her genetic background is like saying somebody is probably a criminal based on his being black. You can expect counterarguments like
People struggle to get Bayesian probabilistic arguments to work in courts where many bias avoiding mechanisms are already employed. In politics, even the most obvious everyday rational arguments don't work regularly. To expect a fairly abstract argument based on numerical probabilities obtained by a method which almost nobody understands to be persuasive in a heated political debate is naïve.
I wrote an article for h+ predicting that the rapid fall in the cost of gene sequencing will allow U.S. voters to learn much about presidential candidates' DNA. The candidates won't be able to stop this because:
DNA analysis has a decent chance of reducing political bias by providing objective information about candidates. If, for example, 70% of the variation in human intelligence is determined by identified genes then DNA analysis would reduce disagreements among informed voters over a candidate's intelligence.