Dumbledore is a little full of himself
Like, I read Tales of Beedle the Bard, and I was struck by how Dumbledore comments on his own cleverness and knowledge in his notes incredibly often:
This prejudice eventually died out in the face of overwhelming evidence that some of the world’s most brilliant wizards(3) were, to use the common phrase, “Muggle-lovers”.
[...]
3 Such as myself.
(Albus Dumbledore on “The Wizard and the Hopping Pot”)
I think I may say, without vanity, that both my Fountain and my Hill performed the parts allotted to them with simple goodwill. Alas, that the same could not be said of the rest of the cast.
(Albus Dumbledore on “The Fountain of Fair Fortune”)
Even I, Albus Dumbledore, would find it easiest to refuse the
Invisibility Cloak; which only goes to show that, clever as I am, I remain just as big a fool as anyone else.
(Albus Dumbledore on “The Tale of the Three Brothers”)
The guy can hardly talk about anything without talking about how smart and wise and brilliant he is. Like, no humility whatsoever.
In the books, everyone keeps singing his praises like Dumbledore can do no wrong and the only one who keeps saying Dumbledore can be wrong is Harry. And even then, in Harry's limbo vision of King's Cross, which I don't think is really Dumbledore, it's telling Harry envisions him saying something like this:
“And you knew this? You knew — all along?”
“I guessed. But my guesses have usually been good,” said Dumbledore happily
Dumbledore doesn't speak to Harry all that often throughout the series, with book 6 being the one he interacts with him the most. And we see that even in conversations with people, Dumbledore loves to hear how wise and great he is. When he says "I might be mistaken" it's with the tone of "I'm right and everyone else is wrong". Which is usually the case often enough, yes (though not always), but he does it a lot, and I found it interesting how often he uses this phrasing and how smug he seems about it:
And then Dumbledore called out from the back row where he stood with the other teachers — “Aha! Unless I am very much mistaken, the delegation from Beauxbatons approaches!” (GOF)
“I may be wrong,” said Dumbledore pleasantly, “but I am sure that under the Wizengamot Charter of Rights, the accused has the right to present witnesses for his or her case? Isn’t that the policy of the Department of Magical Law Enforcement, Madam Bones?” he continued, addressing the witch in the monocle. (OotP)
“Payment?” said Harry. “You’ve got to give the door something?”
“Yes,” said Dumbledore. “Blood, if I am not much mistaken.” (HBP)
Dumbledore uses this phrasing when he knows what he is saying is correct. He is saying it not because he thinks he might actually be wrong. When he actually thinks he is wrong, he makes excuses and tries to reason why the decision he made was actually reasonable at the time:
“Harry, I owe you an explanation,” said Dumbledore. “An explanation of an old man’s mistakes. For I see now that what I have done, and not done, with regard to you, bears all the hallmarks of the failings of age. Youth cannot know how age thinks and feels. But old men are guilty if they forget what it was to be young ... and I seem to have forgotten lately...”
He is incapable of saying: "I was wrong, it happens, let's move on," it has to come with reasoning or an excuse. He blames it on his age, not that he made a wrong judgment call. This isn't humbleness.
Dumbledore is a character who wants to be humble but just isn't. he considers modesty a virtue. Hell, humility is practically his favorite trait Harry possess:
Harry, who could not see any way out of this without flatly lying, nodded but still said nothing. Slughorn beamed at him.
“So modest, so modest, no wonder Dumbledore is so fond — you were there, then?
(HBP) - Slughorn mentions how Dumbledore appreciates modesty.
The third brother in the story (“the humblest and also the wisest”) is the only one who understands that, having narrowly escaped Death once, the best he can hope for is to postpone their next meeting for as long as possible.
(Albus Dumbledore on “The Tale of the Three Brothers”)
He appreciates being humble and modest and sees it as being wise. He derides Tom for thinking of himself as "special" or "clever" even when it's true (and when he does the same). He loves Harry's modesty, which is really low self-esteem, not modesty. Harry's low self-worth is like the ultimate humbleness in Dumbledore's eyes because he doesn't see it for what it is and he was never humble in his life, so he doesn't really know where the balance between confidence and arrogance is or the line between modesty and low self-worth. I think he honestly doesn't know because he is exceptionally arrogant.
Dumbledore created this image of ineffability around him and it's clear Harry is one of the only people (besides Dumbledore and Aberforth) who knows Dumbledore can make a mistake and he keeps reminding Hermione, Lupin, and literally everyone else of that fact:
“People have said it, many times. It comes down to whether or not you trust Dumbledore’s judgment. I do; therefore, I trust Severus.”
“But Dumbledore can make mistakes,” argued Harry. “He says it himself. And you” — he looked Lupin straight in the eye — “do you honestly like Snape?”
This is all another case of Dumbledore being incapable of practicing what he preaches. He values modesty, but he doesn't seem to be capable of it.
Now, I'm not saying he isn't clever or special, he is. But he is the type of really smart person who looks down on anyone they don't see as intelligent as them. He doesn't see most people as equal to him.
Dumbledore doesn't see most of the Order or Aberforth as his equals. He never did. Elphias Doge kisses his ass, but Dumbledore clearly doesn't share the same level of respect for him. Or for most people, really.
“Elphias Doge mentioned her to us,” said Harry, trying to spare Hermione.
“That old berk,” muttered Aberforth, taking another swig of mead. “Thought the sun shone out of my brother’s every office, he did. Well, so did plenty of people, you three included, by the looks of it.”
Harry kept quiet. He did not want to express the doubts and uncertainties about Dumbledore that had riddled him for months now.
[...]
“Grindelwald. And at last, my brother had an equal to talk to someone just as bright and talented as he was. And looking after Ariana took a backseat then, while they were hatching all their plans for a new Wizarding order and looking for Hallows, and whatever else it was they were so interested in.
Dumbledore doesn't trust the majority of the Order with anything because he doesn't think they'd be capable of handling it because they're not him. He literally tells them nothing until he has to, keeping them busy guarding a prophecy he knows can't be stolen by a run-of-the-mill Death Eater. He only tells Harry about the Horcruxes because he has no choice but to tell him. Same with Snape — Dumbledore trusts him out of necessity.
Snape and Grindelwald are the only people we see Dumbledore show respect towards their abilities, wisdom, and magic in some capacity.
Like, he calls Sirius clever, but he talks about him as foolish in the same breath. He calls McGonagall wise, but he clearly doesn't think she's wise enough to be told anything or trusted with anything. And while he does speak highly of Harry's courage and humility and though Harry is insanely powerful and with the right training could beat Dumbledore, Dumbledore keeps putting him down when it comes to magical abilities/intelligence compared to himself:
“I’m not upset.”
“Harry, you were never a good Occlumens — ”
(HBP) - even though Harry can and does get really good at it once he does it his way.
“I do not think you will count, Harry: You are underage and unqualified. Voldemort would never have expected a sixteen-year-old to reach this place: I think it unlikely that your powers will register compared to mine.”
I find this tendency of Dumbledore to be really interesting. He underestimates people constantly and thinks too highly of himself. and he is very honest about it to people's faces. He keeps talking about how Voldemort’s defenses on his Horcruxes are shit, and how Voldemort is foolish when the curse Voldemort left on the ring is literally killing him at that very moment:
“I do not think you will count, Harry: You are underage and unqualified. Voldemort would never have expected a sixteen-year-old to reach this place: I think it unlikely that your powers will register compared to mine.”
These words did nothing to raise Harry’s morale; perhaps Dumbledore knew it, for he added, “Voldemort’s mistake, Harry, Voldemort’s mistake ... Age is foolish and forgetful when it underestimates youth. ... Now, you first this time, and be careful not to touch the water.”
Dumbledore thinking himself so clever, more clever than Voldemort, is what killed him. His arrogant insistence that he's the smartest man in the room killed him. He is undermining Voldemort for mistakes similar to the ones he makes regularly when interacting with Harry. And he's aware of that. He knows he's a hypocrite:
When I discovered it, after all those years, buried in the abandoned home of the Gaunts—the Hallow I had craved most of all, though in my youth I had wanted it for very different reasons—I lost my head, Harry. I quite forgot that it was now a Horcrux, that the ring was sure to carry a curse. I picked it up, and I put it on, and for a second I imagined that I was about to see Ariana, and my mother, and my father, and to tell them how very, very sorry I was . . .
“I was such a fool, Harry. After all those years I had learned nothing. I was unworthy to unite the Deadly Hallows. I had proved it time and again, and here was the final proof.”
I think Dumbledore's self-awareness is why he wants to like Harry as much as he does. While I don't think Dumbledore knows Harry as well as he thinks he does, what Dumbledore does see is enough for him to imagine Harry in his head as this perfect, virtuous martyr that he wished all his life to portray himself as. He idealizes who he imagines Harry is without fully respecting Harry as his own person with his own abilities.
I just find it interesting that for a character who speaks so highly of humility, he doesn't seem to possess it, and that it ends up being the death of him.