You just said it. The base case is true. The inductive case is wrong. You called it true “with sleight of hand”, but that’s more of a poetic statement; mathematically it’s simply wrong.

One could write a completely different proof where the base case(s) cover n=1,2 and the inductive step is as in the post. In such a proof, the base case would be wrong and the inductive step would be correct. But that’s a different proof, not the one in the post.