Is the Schrödinger equation contingent or tautological?

I have been wondering whether Schrödinger partial differential equation of quantum mechanics can be used to model the uncertainty in a dynamical system, say a control system or some estimation process. So the question is whether following the equation is a special property of a particle, or a general model of uncertainty increasing over time.

Now of course the fact that an electron obeys the Schrödinger equation does say some about its nature - it is a falsifiable statement which happens never to have been falsified. But perhaps the left hand side of the equation can be considered, roughly speaking, as a truism: "look, if an object has a probability distribution that spreads ou over time, this is how it should be described." The only physical statement would be that the description applies to an electron, or another particle, not in the equation itself.

The question has nothing to do with whether some philosophical version of quantum mechanics such as the Copenhagen interpretation is correct. It is a pragmatic question: can I use wave functions to represent uncertainty in a practical application? If so, how? An example would settle that the methods of quantum physics can be used as a general purpose tool, not that they have any special physical meaning. The main reason to doubt that the method would work is that the probability distribution obtained as the magnitude squared of the equation's solution has a wavelike character, which is probably not shared by an arbitrary control system.