It’s quite simple. Voting is irrational.
This depends on a couple of assumptions:
The cost of voting is greater than the benefit you receive, including whatever good feelings you get from doing your duty, supporting the good guys, etc.
The cost still outweighs the benefit even after taking into account the total expected benefit to other people, multiplied by however much you care about this.
For someone who feels good about voting, it can be a rational thing to do even if the probability of affecting the result is negligible or zero. And for someone who finds voting annoying but cares a significant amount about others who will be affected by the result, it’s entirely possible for voting to be rational. Generally, the smaller the probability of one vote affecting the result, the greater the number of people who will potentially be affected by it, so these factors can balance out even in very large elections. (You may argue that there are higher-impact ways to be altruistic, which is probably true but doesn’t necessarily matter; usually the choice isn’t “vote xor make an effective donation”, it’s simply “vote xor don’t bother voting”.)
(I know you went on to talk about the possibility of voting as “a charitable or recreational activity”, and I know the main point was to describe why people won’t bother becoming informed voters. But I still think it’s worth pointing out that your opening claim is far from obviously true.)
In the case of Calibration Trivia, my gut reaction is that you’re being a bit unfair to the ‘clever fellow’ (or at least to the hypothetical version of him in my head, who isn’t simply being a smartarse). It sounds like you’re presenting Calibration Trivia as a competitive game, and within that frame it makes sense to poke at edge cases in the rules and either exploit them or, if the exploit would clearly just be tedious and pointless, suggest that the rules are preemptively tweaked to unbreak the game. I know the ultimate purpose of the game is to train a real skill, but still, you’ve chosen gamification as your route to that goal, and maybe there are no free lunches on offer here; to the extent that people derive extra motivation from the competitive element, they’re also going to be focused on the proxy goal of scoring points rather than purely on the underlying goal of training the skill.