As someone who lives in an area with plenty of high-density residential buildings, I can attest that I love it and I wish there were more American cities that were an option for my lifestyle.
What do you mean? In my experience outside of the few major cities (New York, LA, etc.) there are only high rises in the very downtown of large cities.
I guess I tend to think of "high-rise" as a synonym for skyscraper. Something in the neighborhood of 10+ stories. I'm fairly certain that Columbus at least doesn't have many residential buildings that high outside of its downtown.
However, if we are counting 3 or so story apartments then there are definitely high-rises all over the place like you say.