However perhaps this is an uncomfortable truth, that we may or may not choose to face.
take a sip at what educators are trying to push nowadays (not the [majority of] business paraphernalia from the ed. tech world)... it's usually about teaching kids types of skills not found at the grade; like emotional intelligence, resilience and even meta-analysis of their way of studying!
"Good at one thing" in children generally implies "parents are involved" which generally implies "better than average socioeconomic status" which generally implies "children at least relatively decent at most educational things". This is going to be doubly true for things like art or music (or anything with private coaching) which require a significant monetary outlay in order to be good.
About the only place where this doesn't hold are the extremes where high spectrum people can function spectacularly in math and science and yet be ferociously terrible outside of that.
The most fundamental way in which parents are involved in the life of their children is through their gametes. Unsurprisingly smart kids almost always have smart parents, which is a double advantage.
Let me explain. Doing a "simple" study like this is nigh-impossible within the science establishment we have today. Like you said, let me try. Let's walk through the steps.
A PI would have to get a grant to help not the underserved/underprivileged kids, but rather those who are perceived to have an advantage. They have to work with hundreds of schools (gifted kids are rare). Within that set of hundreds of schools they have to control for cultural and socioeconomic backgrounds and so on. Having succeeded they'll have to get this published (and probably fight some accusations of being elitist in the process). All I am saying is: this PI better already have tenure.
This problem is not just about gifted kids. A commonly heard complaint in the field of psychiatry for example is that we only study the minds with pathologies. We don't at all look at how to improve things for healthy people. Come to think of it, that is true in medicine in general.
You are left with comparisons between school districts and/or states