The issue I'm having with the claims is that their use of the word "model" seems to connote exactly of the opposite of what someone reading the paper without se

The issue I'm having with the claims is that their use of the word "model" seems to connote exactly of the opposite of what someone reading the paper without seeing their definition would suggest, and indeed what other commenters here are taking it to mean. 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.