No, the model doesn't need to be "absolutely spot on" for it to be correct about the correlation between industrial activity and sustained global temperature rises.

And that argument works better in reverse anyway: we're talking about a trend implying large parts of the planet becoming uninhabitable for present levels of human population within a relatively short time frame (which will also have an effect on economic activity!). If contrarians really have a good faith belief that this trend is caused by some other factor and will be self correcting, perhaps they could deign to specify their own testable assumptions and make their own model...