For instance, a standard thermostat keeps your house at 20 +/- 2 degrees C. In order to keep it at 20.00000000 exactly, it might need to model every air molecule bouncing around.
The colloquial restatement of the theorem suggests a regulator "is a model of [a] system" if the regulator's action is a function of the system, and nothing else. As in, a constant regulator that has only one state still meets the criteria for being a "model". The possibility that is excluded is that the regulator has additional details that don't correspond to states of the system.
In your thermostat example, the theorem doesn't even begin to say that, even though that's what the title suggests, and what everyone wants to believe. In fact, the theorem merely says a thermostat that only depends on the temperature of the room is simpler than one that also cares if the walls are blue.
So, it may be a correct proof, but it's a useless one.
But you didn't need my comment for that, the fact that there is an entire engineering discipline thriving on doing exactly what the article claims to be impossible should suffice.