If you think that culture matters but institutions do not, look at North and South Korea. If you think that culture does not matter at all, look at differences among different ethnic groups within countries.
South Korea/North Korea per capita income ratio: 14. (Source: CIA World Factbook)
American Asian/American black average household income ratio: 1.9. (Source: U.S. Census)
From a global standpoint, it sure looks like economic policy (a term I prefer to the vague “institutions”) matters a lot more than culture. And controlling for IQ would make culture look less significant than it already does, without making a dent in disparity between North and South Korea.