I've read all the Harry Potter books before, though not in a long time. Lately, I've been listening to the audiobooks (Jim Dale narration), mostly while running, and things have jumped out at me. One point is that Albus Dumbledore may be the greatest wizard of all time, but as a headmaster, he's pretty bad. (Kind of reminds me of Spock's criticism of Kirk's driving: "Captain, you are an excellent starship commander. But as a taxi driver, you leave much to be desired.")
As Headmaster, Dumbledore is in charge of staffing at Hogwarts. (Rita Skeever states that in her article on Hagrid's lineage in Harry Potter and the Goblet of Fire, and while Skeever is a sleazy journalist, there's no indication that she reports falsely.) Let's consider Dumbledore's hires:
1. Quirrel (Defense Against Dark Arts): He isn't a particularly bad instructor, once you put aside the possession by Lord Voldemort, but he doesn't seem terribly effective either. By itself, this wouldn't be a strong indictment against Dumbledore, but it's only the beginning.
2. Gilderoy Lockhart (Defense Against Dark Arts): The usually perceptive Hermione gets taken in at first by Lockhart's "charm," but it should have been obvious that he was a fraud. Indeed, any reasonable interview should have weeded him out - just have the guy do a mock teaching session!
3. Remus Lupin (Defense Against Dark Arts): Lupin was more or less the BEST teacher that Harry Potter's group ever had at Hogwarts. He knew the material, he taught effectively, and he was extremely compassionate and understanding. True, he was also a werewolf, and when unmasked, he resigned. But why did Dumbledore accept the resignation? In Goblet of Fire, he refused to accept Hagrid's resignation, so he certainly could have done the same with Lupin's. And he should have, given that Lupin's reason for resigning was that the Hogwart parents wouldn't have tolerated their children's being taught by a werewolf. In other words, bigotry. I guess Hogwarts does not have a Wizards with Disabilities Act....
4. Rubeus Hagrid (Care and Feeding of Magical Creatures): Was Hagrid even qualified to teach? He was kicked out of Hogwarts in his third year. Strange the lengths that Dumbledore went to in order to keep Hagrid employed, and yet he wouldn't lift a finger to stop Lupin from resigning. Anyway, Hagrid seems pretty incompetent as a teacher, yet he remains on the faculty.
5. Sybil Trelawney (Divination): Up until Harry Potter and the Prisoner of Azkaban, she had made a grand total of one prediction that came true. During Azkaban, she made a second. While in a trance. What's astounding is that Dumbledore knows that Trelawney is a fraud; he's the one who tells Harry that her trance-prediction about Voldermore's servant (Wormtail) was the second ever correct one. Why did Dumbledore continue to waste the time of so many students by allowing Trelawney to keep teaching Divination?
6. Severus Snape (Potions): Ah, finally, Snape. How in good conscience can Dumbledore keep on the faculty such a bully who cheats by consistently discrimination against Gryffindor by taking away points while not doing so when Slytherins engage in the same or worse conduct? Yeah, he's great at potions and he bravely played double agent, while means Snape is a great patriot, but as a teacher, he's pretty bad.
You tell me - how did Dumbledore keep his job for so long?