OK, I guess my biggest complaint is this:

"If this approximation is close enough to the true value, the rest of the argument goes through: given that the sum Δx+Δy+Δz is fixed, it's best to put everything into the charity with the largest partial derivative at (X,Y,Z)."

What does "close enough" mean? I don't see this established anywhere in your post.

I guess one sufficient condition would be that a single charity has the largest partial derivative everywhere in the space of reachable outcomes.