And of course, since you never got a driver's license you don't know what you're missing (and works around it)
E.g.: you have to pay/ask for someone to get something slightly oversized (either a store delivery or get a taxi), and/or doesn't have access to cheaper stores.
Pet transportation is limited.
You're pretty much dependent on others to places not covered by public transportation (and of course, this limits where you can live, and convenient public transport nearby implies a more expensive place).
Of course, for some people this makes no difference, and things like Uber make it easier nowadays.
My local tram network has 2 types of tickets: (a) Adults or (b) Children/pets.
I live in the UK in a small town and can not imagine having kids but no car. Most of my friends find having a car is essential even without kids (though I prefer to not have a car unless 100% necessary).
On my own I am fine with public transport, but anything between walking around the village or going to a big city isn't really covered by public transport.
Anyway, for day-to-day travel and commuting, even with children, cars are not required in a medium sized city like Edinburgh.
One interesting (albeit off-topic) point she and her husband made to me was that having kids in push-chairs/buggies suddenly makes you appreciate the issues that disabled people in wheelchairs must have all the time for access to buildings.
But essentially depending upon others to get out of the city for the weekend to anyplace that isn't likely an other city? Or being able to travel to destinations outside of towns/cities without having someone along who can drive? I'd find it incredibly limiting. In fact, I'm in the midst of planning a couple of hiking weeks in the Pacific Northwest that would be utterly undoable without renting a car.
Whatever works for you of course. I just can't imagine it.
Sounds like a mostly American perspective to me. I'm Norwegian, living in the UK, with a 6 year old son. I did consider getting a drivers license (never felt the need for one before) when we had our son, and it was a bit tiresome when he was too little to walk, but I don't feel a need for it any more. Maybe I'll get a drivers license one day, but I don't bet on it (I'm 40 and have done just fine so far)
Certainly there are plenty of people in Europe and elsewhere who see a car as a nice convenience that becomes even nicer to have when you get a kid, but I don't know many people that see it as a necessity.