Well one definition of "natural" could be that it "came from nature", and as the Schrödinger equation perfectly describes non-relativistic events in nature arguably it came from nature just as much as newtons laws of motion.

Another measure of naturalness is "simplicity", and the Schrödinger equation is a fairly simple expression.

You might argue that it's results don't feel very intuitive, and therefore it is not natural. But then I why you think its intuitive that levers work.