My interpretation of control is fixing all other variables (the ones they mentioned) except for the one being measured (political orientation). If that's not what they did I'm happy to learn.
In that case I don't understand your original comment, as it describes the sort of mistake that arises when you don't control for other factors, but you appear to accept that they did.
> So if I just assign the majority label to all of the population of a given demographics group, I would get the same result right? i.e., predicting "left" for all minorities under 30. You would also get ~70% accuracy.
I meant that even if you control age, gender, ethnicity, a very trivial predictor (i.e., always predicting the majority label) could yield similar performance. What I meant to say was that their model may not perform as well as they made it sound.