Having lived in Japan, I get what you mean about the American suburbs sounding claustrophobic, but now that I live in a pretty typical, American suburban subdivision, ironically a lot of my neighbors would feel the same way about the idea of living in a dense city.
In terms of walking for leisure and exercise, some do, but not everyone. I see people walking/jogging/biking through the neighborhood throughout the day. But the rate at which people do is likely lower than those living in more urban areas. As mentioned by commenters above, just about the only reason to walk outside in the suburbs is leisure/exercise. The nearest non-residential area from me is about 1 mile away, and that's the city core, with some restaurants, dental offices, etc. The closest grocery is probably another half mile. And I'm far closer than most. I'd say the average distance to a non-residential area for most of the city's residents is closer to 2 miles, and the average distance to a grocery store is probably near 3.
But even strictly for leisure, there are likely fewer people walking here than other cities/countries. When I lived in Japan, I went walking because if you go outside, you are basically in the street. It incentivizes walking somewhere like a park, or a shop, or path, or somewhere interesting. In a neighborhood like mine, most people have large enough yards where they can easily relax there. You can throw a football or kick a soccer ball easily. Houses with kids will often have swing sets or trampolines in their backyards. I'm in Indiana, so I estimate no less than 40% of households have a basketball hoop in their driveway or on the street in front of their house. A lot of the things that drive you to walk places in denser cities simply don't exist. Walking outside becomes a decision to walk for the sake of walking, because most people already what they would be going to a park for at home (greenery, space, playground/sporting equipment, even pools are common), and you aren't going to be close enough to actually walk to many destinations.
Yes for sure Americans walk for leisure and casual exercise.
It's popular to imply Americans don't walk. It's true the culture is more pragmatic (why walk when I could drive and get more done), but it's not true that it's unusual to see people out walking on sidewalks, parks, and lakes.
My neighborhood has a walking trail around a small lake (takes 45 minutes to do a lap). There are 20-50 people walking it at almost anytime of day.
Just reading this makes me feel kind of claustrophobic.