Cuaron did nothing of the sort. Gambon's Dumbledore in Azkaban was superb. Maybe the only thing it was missing was Harris's super ancient "of the wizard nobility" element, but the rest of it was a bull's eye. In PoA, he was crazy and wonderful. Except for his big speech at commencement, he was quiet. He was kind. And quite jubilant about all the oddities of the magical universe. The entire time-traveling sequence is wonderful for the character. Watch his mischievousness when they've done it, and he says, "did what?" and bounces away humming a tune. Listen to him distract Fudge with his long name, and with his description of the flowers planted by the previous headmaster. Then his casual dismissal of the Minister and the headsman once Buckbeak's gone. He almost doesn't seem as if he'd ever been a student, but instead had lived in the highlands as a nature spirit before Hogwarts was built, and endured all this time.
It was Mike Newell turned him to an angry boarding school headmaster who shoved and pawed at others. This survived into Order of the Phoenix due to Dumbledore's role in alienating Harry. He only started to come back in Half-Blood Prince, which I think is one of the bright spots of that film. His performance in the cave -- the rushed drinking scene aside, and that may be down to editing -- is inspired.
BTW, the backstory about James and company was in PoA. It was written, it was filmed, but cut at the last moment after test screenings complained it made the ending too long.