For nearly essential services (like child care, groceries, etc.) having a population that must go to one or a few particular establishment(s) out of geographical necessity is a recipe for rent seeking and other bad behavior.

See also: food deserts

I have 5 large child care facilities in walking distance (< 15 minutes), countless smaller ones, 4 supermarket chains within 10 minutes plus all the smaller stores and I don't even live in a particular dense area of Berlin. The denser the city gets, the more options can be made available within a certain distance.

There are at least 6 supermarkets within a mile of my apartment (in Dresden, not an incredibly dense city). That's plenty of competition in practice. And many other services like child care or health care are not an issue either since those are (mostly) paid for by the state.

Where I grew up there were three Kindergartens and two supermarkets within walking distance. A short bus ride away were many more. How many options do you think are necessary to avoid the bad behavior that you fear?

Those sound like reasonable numbers. A number of options higher than 1 within walking distance is atypical in my experience.

I'm not sure, are you saying that cars promote diversity and prevent food deserts?

I find the opposite to be true. If there are lots of people walking, there will be lots of walkable services and businesses, so you tend to have a lot of choice.

Ownership of houses is usually diverse in dense walkable cities so it's really easy to just rent space at the ground floor to start a business. If you don't like it, move across the street.

Compare this to car cities. People drive their cars to malls. Malls are very much rent seeking. I've heard that they constantly increase rents of the shops so there would be churn - new shops to "keep it interesting" for the visitors.