I doubt the barrier is actually identifying genes. Most genes that affect intelligence are probably not a clear-cut win for any of the alleles, but are trade-offs for different modes of thinking. Plus of course it's questionable how much of an effect genetic intelligence has on a President's success.
Furthermore, even if we could determine a numerical intelligence from DNA that worked like IQ, but combined all factors into an unambiguous ranking, and these were published on the Internet about every politician ever, I doubt it would affect votes all that much.
If, for example, Sarah Palin had many identified "high IQ genes" I doubt that the mainstreem media would have been able to characterize her as a dullard.
This study claims to "unequivocally confirm that a substantial proportion of individual differences in human intelligence is due to genetic variation."
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.