You are correct. The average American city dweller now experiences a lower crime rate than in suburbia. (That is an average, obviously some cities are dangerous and some suburbs are very safe.)
> Rich people live in the city
Yes, this is the long-run, natural state of cities. It's why they exist in the first place. They are where economic activity is most concentrated.
American has had a weird 60 years where we inverted that situation, but it is already beginning to revert back to historically normal.
SUVs feel safer, and they're sold as a safe choice - even though in practice they're dangerous cars.
In the same way, suburbia has always been sold as a safe option, even though politically, culturally, socially, and ecologically, the effects are devastating.
Nothing about the south side of Chicago being "risky" is an inherent result of being in a city.
As for safe streets and good schools, it's a product of segregationist urban planning, not any intrinsic qualities of cities versus suburbs. We don't build public housing out in the suburbs, we do it in the cities. Our lack of transit makes it impossible for poor people to drive around in the suburbs. So our city schools are 90% low-income and have the problems that come with concentrated poverty.
Note that in places like Paris, the situation is reversed. Rich people live in the city (or out in the country). The suburbs are where the poor are herded into ghettos.