In Philosopher's Stone, Hagrid tells Harry how he was hired by Dumbledore.
"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.”
But in the next book, we learn through Riddle's diary that Dumbledore wasn't the headmaster when Hagrid was expelled, Armando Dippet was. Every indication throughout the books is that headmasters make the hiring decisions, and as Dumbledore states in Order of the Phoenix, only the headmaster can allow people to continue living in Hogwarts:
"As High Inquisitor you have every right to dismiss my teachers. You do not, however, have the authority to send them away from the castle. I am afraid,' he went on, with a courteous little bow, 'that the power to do that still resides with the Headmaster, and it is my wish that Professor Trelawney continue to live at Hogwarts."
Confusing matters even further is that Goblet of Fire implies that when Mrs. Weasley was in school, there was a completely different gamekeeper.
Mrs. Weasley was intrigued by the Whomping Willow, which had been planted after she had left school, and reminisced at length about the gamekeeper before Hagrid, a man called Ogg.
Mrs. Weasley is older than Harry's parents, but surely she can't be THAT old- Hagrid and Voldemort are older than 70. This whole Hagrid timeline just seems completely off. Is this just another famous example of JK Rowling's math ineptitude, or is there another possibility I'm missing?