I agree. I'm not suggesting his behaviour resembles that of dementia patients. I don't, in general, try to predict the events of the story by modeling the characters as people, since they're not. The question I'm trying to answer is "What is this doing in the story?" Whether Dumbledore turned out to be fundamentally sane or insane, the outcome wouldn't be surprising. It wouldn't serve any larger purpose. It has no pedagogical value, no bearing on the story's major themes, nothing to justify the deviation from canon. The actual mystery is not whether he's only pretending to be crazy but why the subject keeps coming up.
This is the only satisfactory answer I've seen so far: that he's pretending to be crazy to conceal advancing senility. It's not, I realize, strongly supported by the text, but it is strongly consonant with what I already believe. My model of Eliezer-as-author says that he wouldn't merely symbolize the cognitive decline of aging as a supernatural process ('Dementation') which steals the memories of its victims. I expect him to hammer the point home. And impassioned speeches on the horror of aging will have a lot more weight in the presence of a character undergoing the real thing.
(And somewhere in the back of his mind was a small, small note of confusion, a sense of something wrong about that story; and it should have been a part of Harry's art to notice that tiny note, but he was distracted. For it is a sad rule that whenever you are most in need of your art as a rationalist, that is when you are most likely to forget it.)
Why does the wizarding world believe Voldemort used the Killing Curse on Harry? Whether or not the Love Shield exists in MoR, I doubt most wizards had an >epsilon prior for the Killing Curse resulting in a scarred but otherwise unharmed target, a dead and burned spellcaster, and a destroyed building. There were no surviving witnesses except Baby Harry. Where did that version of events come from?
If I was Joe Random Wizard and heard that evidence without names attached, I would naively hypothesize: Dark Wizard shows up at house, encounters mother + father + their allies. Battle ensues. Parents and Dark Wizard are slain. House is destroyed and baby is hit by debris. There is one obvious question - why the allies didn't take the baby with them - but any answer to that is more plausible than "There were no allies; the most reliable curse in the world backfired on its most experienced practitioner."
Not that the "reflected curse" story was hard to sell. People are great at not asking the next question when they want to believe.
We have some additional information about the events of that evening:
What really happened at Godric's Hollow?
PS. If the Love Shield does exist in MoR, do you suppose Bellatrix could cast it?