I would suggest that Dumbledore sent Hagrid for Hagrid's benefit.
Hagrid never finished his education and was not a qualified wizard. It would be very easy for him to be marginalized in the Wizarding world. Dumbledore, therefore, went out of his way to give Hagrid responsibilities so that Hagrid would feel useful and important.
When Hagrid was expelled and no one else cared about him, Dumbledore arranged for him to become gamekeeper — something useful he could do even without a magical education — and Hagrid expressed his appreciation for this to Harry in the beginning of Philosopher's Stone:
"Oh well — I was at Hogwarts meself but I — er — got expelled, ter tell yeh the truth. In me third year. They snapped me wand in half an' everything. But Dumbledore let me stay on as gamekeeper. Great man, Dumbledore.
Hagrid expands on this in Goblet of Fire when he is hiding in his hut after Rita Skeeter's article about him:
"Dumbledore was the one who stuck up for me after Dad went. Got me the gamekeeper job... trusts people, he does. Gives 'em second chances... tha's what sets him apar' from other Heads, see. He'll accept anyone at Hogwarts, s'long as they've got the talent.Knows people can turn out okay even if their families weren'
Similarly, when Hagrid and Harry go to Gringotts to pick up the Philosopher's Stone, Hagrid says:
As a matter o' fact, I gotta visit Gringotts anyway. Fer Dumbledore. Hogwarts business." Hagrid drew himself up proudly. "He ususally gets me ter do important stuff fer him. Fetchin' you — gettin' things from Gringotts — knows he can trust me, see.
And later when they are actually in Gringott's Hagrid again displays the importance he feels at having this responsibility:
"An' I've also got a letter here from Professor Dumbledore," said Hagrid importantly, throwing out his chest.
These tasks would not require any advanced magic, so Dumbledore assigns them to Hagrid so that Hagrid can feel like he contributes. As there are a limited number of activities in the Wizarding world that can be accomplished without magic, Dumbledore probably takes advantage of such opportunities as much as possible.
In fact, as soon as a teaching position opened up for a subject that would not require much magical prowess, Dumbledore gave it to Hagrid.
We similarly find that Dumbledore gave Hagrid a part in protecting the Philosopher's Stone, and that this too made Hagrid feel really good about himself:
"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."
Hagrid's chest swelled at these last words. Harry and Ron beamed at Hermione.
Indeed, when Dumbledore appointed Hagrid to retrieve Harry from Godric's Hollow, Professor McGonagall questioned his choice, and Dumbledore simply responded that he trusts Hagrid:
"You think it — wise — to trust Hagrid with something as important as this?"
"I would trust Hagrid with my life," said Dumbledore.
And by the end of Half-Blood Prince it looks like McGonagall might have caught on
"Pofessor Dumbledore always valued your views," said Professor McGonagall kindly, "and so do I."