I don't know if a satire necessarily has to be funny (though that depends on your definition of funny - the kind that actually makes you laugh out loud or the kind where you don't necessarily laugh but can still recognise and appreciate its humour?). OP does touch on this by bringing up how Idiocracy concentrates more on its jokes than on making genuinely salient satirical points (which I'd personally agree is a massive strike against the film), After all, a film that cares more about making you laugh than about making you think about the points it's making is arguably a weak example of a satire (though the best films do manage to find the right balance).
This is a long one, but I find it very difficult to say anything half coherent on this topic in short bursts. Sorry about that
Laughing out loud is certainly not a requirement for something to be funny. In satire nor in other genres. I generally prefer stuff that makes me think, but have tremendous respect for things that actually makes me laugh because it happens so rarely and because I'm fascinated by the ability to effectively manifest primal responses or sensetions through art in general. (Laughter, hypnosis, unease, ecctasy etc.)
I agree with your critique of Idiocracy, but let me adjust my initial definition a hair: Because there are different kinds of satire I wouldn't say ridiculing people in power and their flunkies is a required element in satire as much as ridiculing ignorance in general, which just happens to be an inherent element of power and its followers. So a better one would be ridiculing ignorant or deluded people into behaving as if they were functional adults while giving the less ignorant a healthy laugh.
I believe a genuine urge for social change has to lie behind any «true» satire for it to work. Some might think that's a militant take on things but it's how I honestly see it. To the degree you share that viewpoint there's something to be said about satire that actually manages to make people laugh, or at least is funny, as it helps spread change/hope/empowerment through broader walks of life. Everyone likes to laugh and will typically have some interest in checking out whatever is broadly known to make that happen.
This is to me where satire differs from most other works of art as the work is judged by its goals and ambitions outside of, more than just in and of itself. Labeling C&C as satire might be a stretch, but the raw comedy certainly helps with bringing a generally satirical viewpoint, however vague, to the masses, thus succeeding with a satire «agenda» arguably as well as what more on point satire with smaller audiences typically does. I guess this is why I'm usually disappointed with most "real" satire, as it's rarely funny to me, thus looses the bite I feel is needed for it to come across as having an impact and making a difference. I want to feel like what I'm watching somehow matters. A huge part of what makes Brüno funny to me is knowing practically the whole world has been or is going to be exposed to it.
The guys behind Southpark strike me as pretty easy going, laid back, dudes with a great sense of humour, not particularly bothered by politics and "serious" topics. Nevertheless I think their work has great impact and truly makes the world a better place. I also think they are aware of it and that it's an intended effect of what they're aiming for and want to do.
You could say that unlike other kinds of activism, satire actually has a sense of humour, and that's not something to be laughed at