Here's what we know about Dumbledore's backstory: he defeated Grindewald in 1945; he suspected Tom Riddle years before Voldemort arrived on the scene, and he led the fight against him the first time around; he expected Voldemort to return and prepared for it, and is now leading the second fight; he disagrees with many Ministry of Magic policies, and is aware of its incompetence and corruption -- a situation that appears to have existed for a long time. Basically, Dumbledore's a very old man who's spent a large chunk of his life in non-stop scheming, both against a succession of wannabe Dark Lords and against his own government. I think this has really warped his view of the world and of other people. At this point, I don't think Dumbledore is capable of not scheming, not even in his ordinary, everyday life. To him, all actions are potential strategic moves and all people are potential game pieces in the strategy. That's why he can do things like setting up 11-year-old Harry to confront Quirrelmort, and then just twinkle about it later.
There are two moments in CoS that I found telling. The first is when Lucius and Fudge arrive to arrest Hagrid and remove Dumbledore from Hogwarts. Dumbledore's response is "You will find that I will only truly have left the school when none here are loyal to me." The second is toward the end, when Dumbledore talks with Harry after Harry gets out of the Chamber. "You must have shown me real loyalty down in the Chamber. Nothing but that could've called Fawkes to you." That's the very first thing Dumbledore thanks and praises Harry for. Not for rescuing Ginny, or saving the school from the basilisk, or for keeping Voldemort from coming back, but for loyalty.
Dumbledore judges the people he works with based first and foremost on how loyal they are to him. Not because he thinks he's all that, but because, as I said, he views people as game pieces, and you can't have your game pieces acting up, can you? He values his pieces. He wants to advance and protect them. But he doesn't want them running off beyond his sphere of influence and doing their own thing. I think there's something very ambiguous about Dumbledore's habit of seeking out desperate, socially outcast people and doing them one or two huge favors that leave them bound to him for life. Remus, Hagrid and Snape all fit that pattern, and Trelawney and Firenze appear to join the ranks in OOP. It kind of makes me wonder what Dumbledore has done for Fletcher, Moody and Shacklebolt.
The members of the Order appear to have pretty much internalized Dumbledore's view of things. They view him not only as their leader, but as their conscience. Hagrid believes everything Dumbledore believes, and would never question or disobey him. Snape doesn't seem to believe what Dumbledore believes, but still toes the line until the Occlumency lessons in OOP push him beyond his breaking point. In GoF, Snape's most emotionally vulnerable moments are the ones where Fake!Moody suggests that Dumbledore may not trust him. Remus, confessing his sins in the Shrieking Shack in PoA, feels guilty not so much because he endangered lots of innocent people, but because he betrayed Dumbledore's confidence. "Dumbledore says..." is the running refrain on pretty much everyone's lips throughout OOP -- except for Harry and Sirius, whom Dumbledore has effectively abandoned.
Speaking of Sirius, Dumbledore's attitude towards him now begins to make more sense. (For an excellent discussion of Dumbledore's treatment of Sirius, see this post by darkkitten1. No reason for me to rehash her arguments here.) The problem with Sirius is, he's not loyal to Dumbledore at all; he's loyal to Harry. From Dumbledore's point of view, it's as if he's playing wizard chess, and one of the knights suddenly decides that he doesn't care what happens to the king, he's just going to take care of that little pawn on the left. So Dumbledore does the only thing he thinks he can do -- he sticks his recalcitrant knight into a safe, isolated corner of the board and keeps him from making any moves. Perfectly sensible and strategically sound, as long as you don't expect your game pieces to have any pesky emotions or psychological issue that need to be taken into account.
And Dumbledore doesn't expect it. He works to insure the immediate physical safety of the people he's responsible for, but he doesn't allow for the possibility that Harry might be damaged by being raised by bullying bigots who hate him, or that Sirius, post-Azkaban, might be unable to cope with confinement in a place full of little but bad memories, or that Snape might not have the inner resources to effectively teach mental discipline to a kid he hates (and who hates him right back). Dumbledore admits at least part of this error himself in OOP, when he says he says he was wrong to have expected Snape to overcome his feelings. He's wrong about a hell of a lot more than that -- he still doesn't seem to have a clue about what he's really done to Sirius and to Harry -- but I now think that his claim about having come to love Harry is perfectly true. He's come to see Harry as something more than a game piece, and he has no clue how to deal with that. It's probably been decades since he's had to actually treat somebody as a person. His usual routine for personal interactions -- twinkle, offer a lemon drop, say something amusingly nonsensical -- doesn't cover Harry's situation. So Dumbledore ends up avoiding Harry for most of the year.
None of this speculation has exactly raised my opinion of Dumbledore -- I still think he's a callous and manipulative old coot, however admirable his ultimate goals are -- but it did give me a more internally consistent view of him. His behavior in OOP seemed very out of character to me at first reading, but viewed as the behavior of a master manipulator who's finally coming to feel bad about one of his manipulations (but who's still now willing to let go of the habit), I think it begins to make sense.