It seems to me that Giles and Buffy's disagreement is entirely character driven... assuming there's no backstory we've yet to be told. He thinks that Buffy is pure and heroic but also naive and unwilling to make the hard decisions. So he makes them for her.

"And sooner or later, Glory will re-emerge and make Buffy pay for that mercy, and the world with her. Buffy even knows that, and still she wouldn't take a human life. Because she's a hero, you see. She's not like us."

"Takes more than rousing speeches to lead, Buffy. If you're going to be a general, you need to make the proper decisions, regardless of cost."

"I understand your anger... but please believe me when I tell you we did what we did for the good of all -- "
"He's alive, Giles. Spike's alive. Wood failed."
"... That changes nothing. What I said to you before is still true. You need to learn -- "
"No. I think you've taught me everything I need to know."

Buffy, meanwhile, is trying to grow into her new responsibilities, and feeling the strain - but her sense of duty would never let her set them aside. She's responsible for the new Slayers.... yes, even for Gigi. But Giles won't let her grow up.

Trivia:

I know that a metallic blade can't go into a stone
But you don't have Slayer strength, and you aren't as outright furious as Gigi is right now. True, the axe might break rather than cut the stone - but they use metal blades to cut stone in quarries, don't they? Besides, the axe might be enchanted.

Buffy doesn't strike me as a girl who reads Burgess or watches Kubrick.
When she told Tara she'd watched the Disney film of 'Hunchback of Notre Dame' instead of reading the actual novel Tara believed her, but she was joking then... and she was the only one to know about William S Burroughs' life story too. I think Buffy is rather more erudite than a lot of people give her credit for... the dumb blonde routine is an act she puts on sometimes.