Well, I grew up in southern Illinois, and we had beaver in my literal backyard. Which, granted, was a wetland which flooded several months of the year, but that was mostly owing to the Army Corps of Engineers doing their job, not the beaver (they flooded where we were to prevent flooding elsewhere).

So, unless this beaver eradication program is a 21st century novelty, I can say that it is not generally true of the US midwest.