at least "What is the European Union?" when voting for brexit would be enough.

That's a good (and presumably unintentional) demonstration of the problem - the European Union is a hugely complex, very political beast and the right answer to that question depends heavily on your political frame of reference and priorities. Even just the free trade-related aspects can be accurately described either as giving up control of our own laws and regulations to a centralized bureaucracy or enabling trade by removing non-tariff barriers depending on how you want to frame them - since almost all non-tariff barriers other than differences in regulations are banned by WTO rules they're the same thing, one description just emphasises the benefits and the other the downsides. (I've even seen a certain FT writer who insists he's not pro-EU use both framings - the positive one when discussing the EU and the negative one when discussing post-Brexit trade deals outside the EU.)

True, but maybe there could be some sort of democratic consensus system to establish the questions (one which by design avoids the superficial circularity).

Ask questions that a 5th grader could answer. The level does not hove too be that high for it to be effective.

Or... just ask science questions: "Is planet Earth flat?".

That just tests the ability to stick to consensus - what happens when you get the equivalent of "does the Sun orbit the Earth" in a society before Copernicus and Galileo? You also move away entirely from the original point of the questions - someone with scientific knowledge doesn't necessarily know anything about society and politics. They're separate worlds.

In a society before Copernicus and Galileo that question would have been cutting edge science. So just stick to well understood things.

Even that is apparently fraught because a few people (though no sects I'm aware of) profess belief in a flat Earth for religious reasons. For example, Biblical literalists interpreting Job 38 as implying a flat Earth ("Where wast thou when I laid the foundations of the earth? [...] Whereupon are the foundations thereof fastened? or who laid the corner stone thereof").

That's exactly the point: exclude those people. The idea is not to have everybody pass the test.