Environmental factors are inherently much more difficult to study in social science than genetic factors are, because causality does not imply correlation due to robust violations of the faithfulness assumption. Basically, much of the environment that we are interested in studying are environmental interventions that can be applied to improve things, but people are already trying to improve things, which likely leads to all sorts of correlations between the application of the environmental interventions and the problems that exist, so as to cancel out the correlation induced by the causal effect. Intuitively, you can think of this as "going to the hospital correlates with dying, but much of this is because hospitals have the option of preventing death, not just because hospitals cause death" effect.
Behavior genetics works best when everything is linear and interaction effects are negligible. Two major things to watch out for are what I'd call AxC (aka gxe or gene-environment interaction) and AxA (aka epistasis).
AxC refers to the case where if your child is above average in a genetic trait, then they benefit from being raised in by one kind of family, while if they are below average in a genetic trait, then they benefit from being raised by the opposite kind of family. For example if you imagine that there are some treatments that would work well for highly-genetically-intelligent kids but wreck havoc on highly unintelligent kids.
Behavior genetics usually cannot model AxC, instead it assumes all AxC to just be A (genetics), ignoring the C component. This seems relevant to your question, since in a sense by positing an innately intelligent kid, you are holding A constant, and that would make it more appropriate to attribute AxC to C than to A.
Even worse is AxA, which refers to the case when you've got two genetic factors that work well together, as well as well with neither, but work badly when you've gotten one or the other. However, that's a rather elaborate and extreme effect, so it is unclear how much of a thing it is. I suspect it might happen a lot due to chaos, which tends to make everything interact, however that could also make it interact with E, leading to AxE, which behaves differently.
For the record, in the presence of AxA, twin studies will be biased against C; they will count AxA as a negative amount of C, plus an extra positive amount of A. On its own, this effect might be fairly mild, but if C is lowish, then due to the point I mentioned in the other comment, that will lead to dramatic biases.
I should probably also mention AxE. Twin studies just attribute AxE to E, so it's if anything biased against genetics, but not in favor of rearing, so whatever.
Behavior genetics squares the effect sizes, which biases it very strongly against smallish effect sizes (where "smallish" includes effect sizes on the order of magnitude 0.3, which would often be considered "large" for social science because it is rare to get effects that large). It is possible for the variance due to shared environment to be small, even though the effect of shared environment is large. See this twitter thread for more detail:
https://www.twitter.com/tailcalled/status/1522876250381066242
you need both tail-end education and tail-end genetics to become a "genius."
Perhaps we should distinguish between being a "genius" like a supersmart person who succeeds at whetever he touches, and being a "genius at X" like achieving extreme results in some specific area.
To use your example, Von Neumann was a genius, but he wasn't a chess grandmaster. He did great things at many areas he tried, but chess wasn't one of them. Was it just a question of priorities? Or was his talent somehow compatible with math and computer science, and incompatible with chess? (Like, I am sure he would be good at chess, but maybe not grandmaster-level good.)
So, back to the topic of genius production. Are we just trying to produce general geniuses, and let them randomly find their field of interest? Or is there a specific area where we want huge progress?
My guess about education would be that is more important to make the information available (separate expert books from pseudoscientific nonsense, provide a free access to the library) rather than school in the usual sense. A tutor would be good, especially someone who can answer the followup questions after reading the books.
OK, trying a bit more of tabooing the "genius". What you need is:
Traditional school provides an ineffective version of the tutoring, textbooks, time dedicated to learning, and peers (for an average student). All things considered, that is quite good (for an average student).
But to achieve a genius level, you probably need to optimize much harder in all directions. A personal tutor, at least once in a while. More advanced literature. A genius at high school age might need to discuss the topic with university students.
The question is, what happens if the potential genius is simply not interested. This may happen quite often, and may be undocumented because of selection effect (we only learn about the potential geniuses who became actual geniuses). How many of them change their minds later? How many switch to another track and become geniuses there? (Perhaps we should support track switching.) How many just never become interested at anything important?
Say you wanted to raise a genius. How would you go about doing that?
Assuming that your starting with an above average intelligence child with a greater likelihood of naturally becoming a "genius", can appropriate environmental/educational interventions substantially increase the child's potential for becoming one?
A couple of anecdotes say yes:
(1) Scott's review of "The Man From The Future" shows that Von Neumann, besides from obvious his obvious genetic advantage, benefited from growing up in a very resourceful family environment with rich exposure to top intellectuals & domain experts from all kinds of fields and private tutors.
(2) Laszlo Polgar's "Raise A Genius" suggests that the key is "early specialization" in a subject chosen under the parent's discretion (as long as the child enjoys the subject) in conjunction with 1:1 tutoring and having access to peers that are "mentally appropriate partners."
The field of behavioral genetic paints a somewhat different picture:
Most if not all psychological traits including intelligence being highly heritable, as long as you don't severely mess up your parenting (eg abuse, malnutrition), the specifics of your parenting strategy won't influence your child's outcome that much. Social/educational interventions just ... aren't as effective as one may expect them to be a priori.
But these two views aren't incompatible; some possibilities include:
Tail-effects in education: Since interventions have to scale, they end up being mediocre to "what could be possible." Perhaps having a Really Good Education (like what Von Neumann had) has disproportionate effects on one's life outcomes/genius-ness in a way that standard social/educational intervention RCTs can't capture.
Tail-effects in genetics: Just like how people with very low-intelligence seem to be fundamentally bounded in terms of the tasks they're capable of (regardless of the education they receive), a similar effect might be going on with normal intelligence folks where there's rapidly diminishing return to improved education—i.e. education scales better with individuals genetically predisposed to be smarter.
Both (1) and (2), where geniusness is a "leaky pipeline" in which you need both tail-end education and tail-end genetics to become a "genius."
What is the actual underlying dynamic of "genius production"? My hunch is that things are closer to (3), considering how Von-Neumann-type intellect is much rarer compared to the number of individuals that grew up having access to his level of resources. Also remember, the causal influence is bidirectional.
Also: does there exist prior work on "designing literally the Best Form of Education that's conducive to genius production" (provided that the child is genetically predisposed enough to actually benefit from them)? Longitudinal studies like SMPY comes into mind, but I don't know whether studies like these have been distilled enough to validate the practices of Neumann/Polgar, or produce novel insights for (non-mass-consumption) education.