Who does the best job managing density? Tokyo is lovely and orderly, but it’s not that dense—similar to San Francisco. Maybe Seoul?
They have a population of 6 to 7 million people in an area of 700 square kilometers, resulting in a population density of 8300 people / km^2. Substantially more than that if you account for the fact that a large percentage of the island is still tropical jungle.
Despite that fact, their city planning is so good with large open spaces everywhere interspersed with greenery, that you almost never feel claustrophobic. Even the so-called "hearland" neighbourhoods with rows after rows of high-rise residential HDB buildings are quite pleasant.
The most claustrophobic place I've been in Singapore are the few squares in the center of CBD filled with skyscrapers that almost obscure you the view of the sky.
There is a good argument that San Francisco could and should be denser than it is, but its ludicrous to call it not dense at all.
quick funny story, my family and i were in Times Square last year for New Year's. Thousands of people everywhere as you can imagine. We're walking down the sidewalk and right as rain my wife runs into someone she knows from all the way back in Texas. Among all those people from all over the world she still manages to run into someone she knows. My wife and her talk while me and the boys hang around waiting just like we've had to do at our local grocery store back home. My kids and I still laugh at that story.