Nobody serious is saying that environment doesn't play some role.
'Some role' is very vague, and many do have the impression that the race you're born as is the stronger predisposition rather than being birthed as a byproduct of gregarious trends and survival needs.
Take a race and put them into a drastically different environment, then give them time to produce offspring for enough generations. They will conform to the conditions of their environment overtime as per Darwin, seeing "the fittest" as whatever said environment needs, rather than making it plainly about short term adaptations.
If you live somewhere where brute force power or stamina is respected, then those will be who make more children, while if you're somewhere that prides intelligence or cunning you'll see different gregarious trends turn out a different set of parameters. Even something as small as who the current Pop Idols are can affect who ends up having children vs not based on how closely they can resemble their figures of success.
You acknowledge that genes are largely responsible for outcomes, just not with the conclusion that race (which is defined by clusters of genetic similarity) is a major factor. Or maybe I am reading between the lines too much there? Millenia of different selective pressures and genetic drift within isolated populations are what makes races what they are, in a literal sense. I'm not saying that going forward, each race or ethnic group has traits that are set in stone. But I am saying that it would be naive to think races are just simply visually distinct, and that the internals are not as different as the physiogomy.
Do you believe that someone can become more intelligent than they already are?
That's not my point at all, you can't just throw people who were fit for one environment into another and expect it to work. If that worked, all we'd need to do to fix the problem is throw enough money at it. Rather than a Psychology situation, this is more Sociological.
It's not about someone, but rather how a community can evolve based on what is seen as attractive and capable of survival within those conditions. Control the conditions one needs to be seen as successful hard enough to pass the ingrained history (ie: adoption) and you'll see trends change as quickly as three generations.
Fixing something like this takes the use of media propaganda to reshape what success looks like. I believe the answer, short of drastic reshaping of someone's environment, lies in Cultivation Theory. If you can control what's culturally trending, you can affect who will and won't succeed in producing offspring.
There's also other factors to take into account like quality of sleep and diet which affect the end product, but that's almost a tangent.
If most people in a group did not breed, only those who met some criteria? Then yes, we would see a strong trend toward some direction, relative to the origin. The verbal IQ of Ashkenazi Jews is I think the most obvious reprentative of this, where a group had a genetic bottleneck, and the successful ones at the time were travelling merchants. And because of the history of Jews in Europe, the selective pressure never totally eased.
Ashkenazi verbal IQ could probably be dropped in 3 generations if only the really dumb ones reproduced...but in reality, the modern era has soft pressures. So what I mean to say is it would take another genetic bottleneck, not a change of pop stars. Otherwise, the needle has to be moved slowly and consciously over time. It seems based on what you are saying that you are aware that race differences are more than skin deep, but with emphasis on the caveat that traits are not immutable.