I take the bus and bike to work. Small groceries are also done by bicycle. Whenever I go out to eat with my wife we use a ride sharing service because we enjoy a drink or two with our meals.
Not necessarily disputing you, just wanted some clarification on why you think that. Anecdotally I know several carless couples with children ostensibly making it work for "unplanned" auto trips via Uber. They live close enough to walk for "planned" trips like grocery store/work etc.
Obviously you'd need to live close to those things but that was a major factor in their choice of home (location).
It seems there's spotty availability in a few cities, but the standard from Uber is you get a single car seat appropriate for a child >12 months old.
Or do you supply your own, and then drag the seat(s) around with you at your destination?
Once you get out of car seats and into booster seats, I could see it as pretty workable. Booster seats aren't all that big and are simple to install. But in CA kids need to be at least 8 years old and over 40 pounds before you can use a booster seat.