This largely depends on where you are in the US. I have moved a lot over the years, from dense city centers, to the dirt roads of rural America. Here are my experiences:
NYC would probably be the most comparable to your experience in London, but seeing as I haven’t lived there, I can instead talk about Boston. When I lived inside the subway range in Boston (Somerville specifically), my experience matches up with yours. I was ~5 minutes from a supermarket and ~15 minutes from the subway/train stop by foot. I was even closer to a couple bus stops for lines that would take me to places like a mall, nearby universities, or the next subway line over (we don’t have an equivalent of the Circle line).
I currently live in Boston suburbs (Metrowest for people that know the area) and can’t really walk anywhere as my street and adjacent streets don’t have sidewalks. I could try to walk on the street, but with the narrowness combined with the speed at which people drive through this neighborhood, it would not be fun. If I hop in the car, I am ~5 minutes from a strip mall with a supermarket, pharmacy, liquor store, etc. and ~10 minutes from the commuter rail train station that I use to commute to the city for work. If I want to head to a large shopping hub with a mall, then it is ~20 minutes away by car. There is a skeleton of a bus system in my area, but it would require traversing ~1.5 miles on streets without sidewalks to get to the nearest stop for me.
When I lived in a rural area (rural PA), things were very different. To get to the nearest supermarket (a WalMart), it took ~30 minutes worth of driving. If I wanted to go to the mall, it was closer to 60 minutes. I am sure there are even less dense areas than that in this country.
Boston and New York are probably the closest to European cities with respect to transit that you will find in the States. Plenty of people live in those cities without cars. I lived in Boston for a long time (now in the Boston suburbs) and plenty of adults I know there haven’t even bothered getting their driver’s license.
The other Northeast Corridor cities are probably the next tier down. DC has decent transit if you make live and work near transit stops. Philadelphia can work, but SEPTA has been unreliable at best in my personal experience. I haven’t really spent much time in Baltimore to be able to say.
Outside of the Northeast Corridor, the only other option really would be the San Francisco bay area with its BART system. It has decent coverage and I have family that lives in the area and enjoys it. However, I don’t have much firsthand experience with it.