Well, or I grew up in India and walked daily through poverty that most Americans never see or experience in my blind privilege. But it's really hard to tell exactly what life experiences a total stranger has had, isn't it? Look, clearly you think that the top article's search technique is less than ideal, and as I said I might even agree with you, although I have absolutely no idea how one would conduct the "broad survey" you suggest and look for outliers, given the general unreliability of things like IQ tests and the dependence of their results on cultural background and education (see e.g. the Flynn effect). You also have to factor in Bayes' theorem. Just what do you think the prevalence of mathematical genius in the general population is? How would you test for it? What is the false positive/false negative rate of your test?
There could be a Ramanujan living in the Amazonian rainforest today, but the only manifestation of their genius might be their enormous insight into the ways of an environment that would kill either of us overnight plus their skill as a tribal shaman. Or they might be tribal outcasts or dead -- there is some evidence that genius comes at the expense of repurposing cortex devoted to social or other brain function. How would you identify such a person in your survey not just as potentially mathematically competent, but as a mathematical genius? I think you are just plain mistaken when you assert that their selection mechanism is fundamentally incorrect -- at the very least it pulls from a group that is highly filtered by many things, one of which is, without question, intelligence and mathematical ability compared to most (but sure, probably not all) of the general population.
We might also agree that what they are planning to look at with regard to their genetics is narrow and stupid IF that is really ALL they are planning to look at (which I doubt). They really need to run a huge battery of tests on the individuals in their group and in various control groups and in first and second degree relatives of the in-group and more, both genetic and the other kind. I suspect that they will simply replicate the findings of many others who have already conducted similar studies, that intelligence and accomplishment (as outliers) have a tendency to regress back to the mean (within Flynn effect amplification that might be associated with "privilege") between generations.
But none of this is a particularly good reason to make the whole thing into some sort of quasi-racist nazi class war (on their part). Yes, their sample is in some sense at least partially self-selected and hence subject to all sorts of biases. OTOH, the group they are looking for is one that is going to be very difficult to identify any other way, and the educational system of the world has never been MORE egalitarian and blind to privilege than it is today, and the world has never been more affluent than it is today both. That isn't to say that we are even particularly close to a long term goal of complete social and economic equality (of opportunity if nothing else, since so far we cannot do anything about the distribution of abilities and yes, I have a brother with Down's syndrome so don't tell me everybody is a genius if only the proletariat would rise up against their capitalist oppressors) but at this point the pool of the most talented tenured faculty in math and physics worldwide has plenty of members who were not "born to privilege".
I'm sure that there are few there who probably DID only get there because of intelligent and wealthy parents as I've met them (although even they aren't stupid or they wouldn't make the cut into Princeton or MIT or for that matter Duke). And I'm certain that our educational system fails to attract some of the brightest into academia because they'd rather make money or because they attend a really shitty school and never have a chance to succeed (although as I said, real genius has been known to transcend THAT barrier time and again). And even though now we pull students and faculty from all over the world, especially from China and India, there are still parts of the world that are strongly underepresented, and sure, it is quite likely due to poverty and lack of opportunity and the fact that 1/3 of the world is still living in the 17th century socially, politically, economically, religiously. That still doesn't mean that the pool of talented tenured faculty is the moral equivalent of a gold miner's pan with most of the dross washed out. The thing they are looking for, however one defines it, is a lot more likely to be found in this sample than anywhere else. There will be a few rocks mixed in, perhaps some fool's gold, perhaps the pan didn't pull dirt from around the mother lode, but it is a lot more efficient to study this group than to wander down the stream bed picking up 7 billion pebbles one at a time and looking to see if it is gold.
I "do" statistics at the professional level and I share your concerns about the design of the study and various sources of bias and confounding effects, and I will note even further that things are far worse than you suggest because they do not appear to have a double blind control group per se so that the entire process is an open invitation to confirmation bias or amplification of statistical accidents into "significant" correlations. One possible outcome of the study is that they don't find any common genetic factor(s) that can be identified as "mathematical genius" that haven't already long since been identified and that aren't as likely as not projections of social inequity as you suggest. Another possible outcome is that they do find something (correlation) but falsely identify it as causal because of the small size of the study, a lack of understanding of mechanism, and the lack of random selection methods including double blind and pulling from a broad pool instead of a small biased one. We might both be able to design a better way of looking for the same thing. But all of those ways would be a lot more expensive, and I think you are way too focused on "privilege" as being the prime determinant for inclusion into the study group and hence the primary confounding factor in the study. Surely they could control for that a variety of ways, just as they may well have plans for a control group and may well expect to use random numbers in various places to confound confirmation bias. Or they may not -- maybe this whole thing IS a racist, classist, genetic determinist conspiracy.
If so, I doubt they will convince many people with their final product, whatever it is. But when you are looking for genetic factors for rare diseases, you don't screen the general population. If you understand Bayes theorem, you even understand why. You go to a small subgroup of the population where the disease has already manifested itself (and ideally, to several generations of first and second degree relatives and to a control population) and look there. This is SOP for genetic studies of rare conditions, and it is as it is for good reasons. It works even if the disease in question isn't genetic per se at all, but the result of a privileged class using lead pipes for their water supply or having the unsavory habit of marrying brother and sister together in the royal class. I fail to see the validity of your primary complaint if one substitutes the term "mathematical genius disease" for e.g. "Crohn's disease" or any other. Sure, a stupidly done study could easily fail, but the method isn't a priori stupid because they don't screen the general population for either one but rather focus on the group of individuals where the condition (for whatever reason) has already manifested itself, and then one merely has to hope that the people who do the work aren't idiots because you can't fix stupid and you can't fix (or sometimes, easily identify) badly done science. At least until somebody comes along and tries to replicate their results or use them as predictors to see if they succeed or fail.