I generally agree with the rest of your post, but this part isn't quite true. Very high crime areas also tend to be low income areas, which means less funding for police, which means less police for the population than what they should have/what you'd expect.
High income areas will typically have more police and less crime.....at least in the northeast where everything is municipal based, and you can isolate these things easily compared to the county-based shared services like much of the country uses.