I would guess ... it obscures the perfectly rational denotation of what Rumsfeld said.

Right, that would make sense, except that the very same people, upon shifting gears and nominally changing topics, suddenly find this remark insightful -- "but ignore this when we go back to mocking Rumsfeld!"

I think the verbal (as opposed to situational) humor comes from a combination of saying the word 'known' and its derivatives lots of times in the same paragraph, using the same kind of structure for consecutive clauses/sentences

Wow, you have got to see Under Siege 2. It has this exchange (from memory):

Bad guy #2: What's that? [...]
Bad guy #1: It's a chemical weapons plant. And we know about it. And they know that we know. But we make-believe that we don't know, and they make-believe that they believe that we don't know, but know that we know. Everybody knows.

And I can't immediately think of a direct way of expressing precisely what Rumsfeld's saying without using the same kind of repetition, and what he's saying will always sound obvious once it's said.

Yes, "damned if you do, damned if you don't" is fun, but ultimately to be avoided by respectable people.

Things that are obvious once thought of, but not before, are often funny when pointed out, especially when pointed out in a direct and pithy way.

Right, but aren't they typically followed by the appreciation of the insight rather than derision of whoever points it out?

Strictly speaking that's true, although for Rumsfeld to avoid the question on that basis is IMO at best pedantic; it's not hard to get an idea of what the reporter is trying to get at, even though their question's ill-phrased.

True, but it's not really Rumsfeld's job to improve reporters' questions. I mean, he might be a Bayesian master if he did, but it's not really to be expected.