Your proposal can't actually test the hypothesis because it can't (reasonably) be done. We don't know how many children were similarly situated and did not turn out like Mozart.
It could be that many musicians with a similar background were/are similarly as good as Mozart, and Mozart's music is especially famous for its cultural appeal and his father's aggressive "marketing" to royalty. There are many reasons one particular musician's musicical legacy can succeed aside from raw skill and creativity, much like the success of a company has a nontrivial luck component.
It could also be that he really was extraordinary above and beyond his family's musical background. Like most discussions about the nature versus nurture of genius, we can't really assert one thing or another.
It could be that many musicians with a similar background were/are similarly as good as Mozart, and Mozart's music is especially famous for its cultural appeal and his father's aggressive "marketing" to royalty. There are many reasons one particular musician's musicical legacy can succeed aside from raw skill and creativity, much like the success of a company has a nontrivial luck component.
It could also be that he really was extraordinary above and beyond his family's musical background. Like most discussions about the nature versus nurture of genius, we can't really assert one thing or another.