Indeed, people begin by assuming that since programmable computers can approximate many classes of patterns and since computers are amazing therefore everything is just an instance of this class.

Were this true, I cannot see why any system wouldn't be a model of any system; ie., alchemy would be possible.

What people forget is that "programming" in the alchemical sense requires a star.

Once you get specific about what properties and processes actually correspond to the allegdeged "complexities" you're talking about, its obvious that no programmable machine will implement them; as obvious as none will turn lead into gold.