What's exactly wrong with data controlled by relevant variables such as education, hours worked, and experience? If you find a combination of variables that reject the hypothesis, maybe you should look further into it, no? Dismissing it because you think it's not objective for some absurd reason is rather unscientific. "Trying all possible combinations of variables" does not disqualify an analysis from being objective--not sure where you got that idea from.
What's not okay is sequentially trying all of the possible analyses and then stopping the moment you find the exact combination of variables that tells you what you had already assumed to be true. Especially so when simple analyses point to A, and you keep adding new variables until you get to B, which is exactly the case here. That is a very well known abuse of statistics. There is a reason all the well known and popular Information Criterions (which measure model quality) are parameterized by the number of parameters in the model.
And while adding control variables isn't per se bad, there are proper precautions to take when doing so, which become exponentially more costly the more you add. Such as segmented sampling, non-linearity transformations, and even controlled experiments. Because these fraudsters have a motive, the model only needs to be as rigorous as necessary to secure their predetermined conclusion. The "keep adding variables" model almost always ends up as a way to lie with statistics.
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?
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
Except by then, the whole analysis becomes a charade. When you try all the possible combinations of variables and then stop the moment you find one that rejects the null hypothesis, your analysis isn't objective... it is predetermined.