Dog-latin magic that runs on Aristotelian physics and enables non-Turing-computable time travel is "logic that could actually work in the real world"?

No, it's a set of premises which happen not to be true in the real world. Logic consists of starting with a set of premises and making conclusions. Obviously Harry Potter is going to arrive at different conclusions about his world, than we will arrive at about ours. That doesn't mean that he is using a special kind of logic called "magic-logic".