‘How is the boy getting here, Dumbledore?’ [Professor McGonagall] eyed his cloak suddenly as though she thought he might be hiding Harry underneath it.
‘Hagrid’s bringing him.’
‘You think it – wise – to trust Hagrid with something as important as this?’
‘I would trust Hagrid with my life,’ said Dumbledore.
‘I’m not saying his heart isn’t in the right place,’ said Professor McGonagall grudgingly, ‘but you can’t pretend he’s not careless.’
Philosopher's Stone - page 16 - Bloomsbury - chapter one, The Boy Who Lived
Even Hermione implies Dumbledore's faith in Hagrid is well known around Hogwarts:
‘Oh, come on, Hagrid, you might not want to tell us, but you do know, you know everything that goes on round here,’ said Hermione in a warm, flattering voice. Hagrid’s beard twitched and they could tell he was smiling. ‘We only wondered who had done the guarding, really.’ Hermione went on. ‘We wondered who Dumbledore had trusted enough to help him, apart from you.’
Philosopher's Stone - page 169 - Bloomsbury - chapter fourteen, Norbert the Norwegian Ridgeback
From eagerly introducing an Acromantula into Hogwarts, to bringing Fluffy onto the school grounds, to hatching Norbert ("The baby") Norwegian Ridgeback and practically burning himself and his hut to the ground as Norbert quickly grows, to bringing Grawp to Hogwarts, to inventing Blast-Ended Skrewts, to appearing intoxicated in front of Harry, Ron, and Hermione ... putting it lightly, it's fair to say Hagrid does not always exercise good judgement. Yet, Dumbledore says he would trust Hagrid with his life.
Why is this? Why does Dumbledore trust Hagrid with his life?
Dumbledore is not infallible and does make mistakes -- I think Dumbledore would be the first to admit this about himself. So, what does canon tell us about why Dumbledore would hold such great trust in Hagrid? I'm interested in a canon-based answer from the books, Pottermore, or J.K. Rowling interviews that would further explain Dumbledore's trust in Hagrid. Answers made within the spirit of canon are also welcome. I am not necessarily asking for examples of Dumbledore entrusting his life to Hagrid, as that question has already been asked (see above), although I'm eager to read any information that explains why Dumbledore trusts Hagrid so implicitly.