You're right that the Brookings article suggests that my quote isn't the case. I wish we had more information on how they controlled for different effects. IIRC BLS etc don't generally control for cost-of-living, and Brookings doesn't discuss the issue except for in the conclusion:
Likewise, the bureau should also consider how demand and supply play out in regional or metropolitan area labor markets, since job search and recruitment often happen locally.
I'd be interested in seeing those results, as I would expect a greater concentration of H1B workers in large metropolitan areas (SF/SV, NYC, Boston), where wages are much higher than the flyover states (where I'd expect less racial/ethnic diversity).