I'm not sure this is enough. If you classify all old white men as Republican you can have 69% accuracy. I'd even argue the more such factors you control the less it means something.
I think you misunderstand what "controlled for" means in this case. The way the controlled version of the test works is that two pictures are presented at the same time. Both subjects will be of the same gender, the same ethnicity, and (approximately) the same age. The goal of the test is to choose which of the subjects pictured is conservative, and which is not. There is no way to obtain greater than 50% accuracy by choosing "old white men", because both pictures will be of "old white men" (or "young black women", or whatever). Something else in the photo is being used to obtain the boost in accuracy: perhaps facial hair, perhaps obesity, perhaps apparent youthfulness, perhaps a guess at sub-ethnicity, perhaps pose---but it has to be something other than "old white man".