Except that's a mischaracterisation. No variables or set of variables would ever 'fully' account for a complex social phenomenon like income from wages and salaries.
And yet, that is what you must do to claim there is no evidence of discrimination. More importantly, they aren't even able to "account" for any of the gap because accounting for it requires demonstrating particular causality, and all they have done is show overlapping correlations which do not favor a non-discrimination account over a discrimination account.
the only certain fact we have, which is that the average pay of women is lower than of men
In what way is that the only certain fact? It's not even the demographic difference with the largest effect size: full time workers make more in wages and salaries than those working part time hours. Forty year olds in the labour market make more than twenty year olds.
It is the only fact about the relation between gender and pay, namely that their is a relation favoring men. Any causal account for that relationship is not a fact but an inference based on a host of assumptions.
The burden of proof is on those claiming their lower pay is not due to their gender.
No, it's not. The burden of proof of those claiming gender discrimination is on the people claiming it.
The most parsimonious explanation for the observed covariance between gender and pay is a causal influence of one on the other and pay cannot cause gender. Any other explanation for it requires positing at leas one and probably many other variables that have particular causal relations to both those variables. Unless compelling evidence of such causal relations is established, the most plausible account is the simpler one of gender impacting pay.
The "not gender" theory requires the non-parsimonious assumptions that the genders differ on other variables
It would be
unalloyed madness to imagine that men and women do not differ from each other, on average, on an
uncountable number of characteristics.
And among the countless ways in which they differ is likely to be the different responses they evoke from their employers related to pay.
Note, for example, that men are taller than women. This is generally not regarded as due to discrimination (except by unhinged whacktivists who think that girls are socialised into being shorter -- note that this isn't satire of some position but an actually sincerely held belief).
You don't just have to assume that gender relates to other variables, you must assume that those particular variables it relates to also causally impact pay and are the only reason gender relates to pay.
But to look at the height difference between men and women and then to imagine that all of the difference is due to discrimination is the same as looking at the wage gap difference between men and women and then to imagine all of the difference is due to discrimination.
First, let's be clear that it is the OP and its supporters claiming that all or near all the wage gap is due to something other than discrimination. I am not saying that all of it is due to discrimination, just that there is no good evidence favoring other explanations for any part of the gap, let alone most or near all of it.
That aside, NO, your analogy is a total fail and a discrimination account for pay is pretty much the direct opposite of what it would be for height. In fact, assuming that their is no pay discrimination is more like assuming that gender is only related to height because women do not try hard enough to be tall. Gender/sex is biological and height is a feature of biology. has a direct biological impact upon height. Thus gender can impact one's height in a biology-to-biology relation within a person's body in just about as direct and parsimonious way possible. A direct effect of genes on height will always be more parsimonious that any other route, including discrimination or any other account involving people's behaviors (such as successful nutrition acquisition). In contrast to height variance, pay determinations are not a biological trait but rather inherentl the product of an employer's mental processes in which they distinguish between employees in deciding who gets what level of pay. IOW, by definition, all pay variance is the product of the mental act of discriminating between options. Thus, all possible models accounting for pay variance share the same presumption that discrimination in the general cognitive sense is happening, and they differ in what factors are assumed to impact those acts of discrimination.
Thus, the outcome to be explained is why are employers making discrimination in pay that related to the gender of the employees? Well, the most parsimonious account is the one that simply presumes only the variables in the question itself, gender and discrimination. The employer is factoring gender into their process of discriminating between pay levels for various employees. Gender directly impacting pay determinations (another word for discrimination) is as direct and parsimonious as it gets, and thus analogous to the biology that defines gender impacting the biology of height. In contast, any alternative account for why gender relates to height requires numerous additional variables and causal steps, just like adding a discrimination mediator to explain gender differences in height.
In sum, discrimination is the fat more parsimonious explanation for pay differences for the same reasons that it would be a far less parsimonious explanation for height differences.
and those other variables are the cause of pay differences. There is nothing wrong with extra assumptions
Yet just because you can use a single term -- 'discrimination' -- does not mean your model is more parsimonious. 'Discrimination' is not a single definable act. You may as well say all the difference is due to 'gender preferences' and leave it at that, as if that explains things parsimoniously.
I explain this above. All pay differences require cognitive acts of discrimination, so all explanations for pay differences make the same assumptions about that and differ in what the assume impacts pay differences. The gender discrimination account assumes that a person's biological manifestations of gender directly impact how the employers discriminate pay levels. Any other account presumes that the biolgical manifestations of gender impact some host of other variables that are directly relevant to job performance (without this latter part, it would still be gender discrimination) , and then those variables causally impact how employers discriminate pay levels. IOW, all alternative accounts are between twice as and many times more complex depending on the number of other variables requires to fully account for the gender - pay covariance (and yes, they must fully account for it (not just via shared variance but established causality), or any leftover variance represents evidence of discrimination).
so long as you can account for extra data that theories without those assumptions cannot.
Of course, it already does. Gender discrimination does not explain why women with young children earn less than child-free women of the same age.
Doubtless, you might say 'women with children are seen as more feminine, and more feminine women are discriminated against more'. Such a just-so explanation invokes another dimension to gendered discrimination (level of perceived femininity), reducing the parsimony of the model.
And yet it is just as theoretically plausible as your alternative and at least as parsimonious that your account. Also, that is not data directly relevant to the gender pay gap.
For example, relevant data would be why do gender pay gaps at time 1 predict the amount of child care related employment reduction for both parents at time 2? The timing rules out the possibility that these employment reductions are the cause of the pay difference, and support the reverse causality that pay differences impact employment reduction choices.
IT is no more a "just-so" explanation than all the non-discrimination accounts. In fact, there is massive evidence, including from controlled experiments) that female attractiveness causally impacts job-related treatment by others including compensation, and that perceived attractiveness declines from 18 to average childbearing years.
This is more of an empirical fact than any a data you have or could offer to support your non-gender account. Incorporating established facts is not adding assumptions but increasing coherence with other knowns.
And note that the just-so explanation can apply (with a little modification) if the exact opposite were true. If women with young children earned more than child-free women, someone might say 'this is because child-free women are seen as unfeminine, and the market punishes them for their lack of gender conformity'.
That is a completely different account, not a slight modification. We could just as easily make up the idea that if women with young kids earned more, someone might say that it because their employer sees them as even more hard working and committed to their job given they still show up with all those extra burdens, or if they take more time off, that employers think more highly of them for having priorities and being good parents (in fact, the coach for the Arizona Cardinals told all his employees he would fire them if they put their job before spending time with their kids, watching their games, plays, etc..)
I can just as easily take all of your preferred alternative variables and make up a story that would account for the opposite result. Being able to do so has no rational bearing on anything.