You'll notice that I've never claimed that new parameters universally make the model worse. All of the common Information Criterions [2][3][4] are numerically capable of improvement of model quality with increases in parameter space...it's just unlikely. Such p-value hunting might give you the p-value you're looking for, but it is very unlikely to improve the model.
Sexism is a hard problem. Throwing variables wantonly at models until sexism disappears isn't doing anything for the problem...it's nothing more than a pseudoscientific way of pretending it doesn't exist.
[0] https://www.skepticalscience.com/urban-heat-island-effect.ht...
[1] http://news.stanford.edu/news/2011/october/urban-heat-island...
[2] https://en.wikipedia.org/wiki/Bayesian_information_criterion
[3] https://en.wikipedia.org/wiki/Deviance_information_criterion
[4] https://en.wikipedia.org/wiki/Akaike_information_criterion
The horseshoe nature of politics will never cease to amuse me, though; thanks for the example.
Also, if a hypothesis is supported by simple studies but falls apart under more complex ones, it might be too simplistic a hypothesis. Almost as if sexism (and sexism guilt-slinging) wasn't an entirely black-and-white problem. Who'd have thought?