1) Minor gerrymandering with questions. According to my academic experience economics is being studied by people more on right wing (especially in economics), while political science seems to attracts more left-wing leaning people. Which one should dominate on the exam paper? (not mentioning even higher ideological disparities between military academies and humanities)
Sure, people could learn for the exam, but with limited effort you could actually eliminate less educated and more lazy part of some electorate, which in your opinion is undesired.
2) Gerrymandering by dealing with threshold. For example, in great oversimplification: in the US poorly educated minorities vote Democrats, above that you have let's call it Bible Belt, then college educated Democrats, and on the whole top are over represented classical liberals (libertarians) who consider Republicans as lesser evil. Pending where you put the cut off line, you may slightly skew whole political composition.
3) People don't like being told that they are idiots. It would be even more humiliating when the test would actually show that in quite clear and hard to refute way.
4) Big part of democracy is not electing bright people, but providing social harmony - people frown at you when you say that masses are on your side and you need to throw country in bloody revolution, when if your support was genuine it would be much simpler just to run in next election. Apparently using check and balances (or deep state in soft version like technocratic civil servants) is a bit more palatable for general population.
[Personal opinion: I'm surprised that other answers were excessively general, with limited amount of pointing out where exactly one could skew such system a bit. Nevertheless, there is clear limit in making the test an overkill, as ones favourite electorate would have to be able to pass it without serious problems. Personally, I'd like for such system to be tested on small scale]