That very much depends on the city. I had no problem getting around Paris/Amsterdam/London/Rome when I was traveling with my friend's family (3 kids). A car would have been a bigger nightmare with traffic and parking and we weren't even staying in the central areas.

But it also depends on how and where you live and work specifically. If I lived in a many of the areas outside of the Périphérique in Paris, not owning a car would probably impact your overall quality of life, but I'd still not want to take it into the city.

Rome should not be on that list IMO. Its public transport is terrible, at least in the centre. Still preferable to having to drive there though, that's for sure. I'm sure it's not true of every part of the city, but I was unpleasantly surprised by inadequate rush hour bus service. Haven't seen buses that packed since my childhood in 1990s Ukraine.

I have 2 kids and live in Berlin car-less without trouble.

We use public transit and bicycles (lots of parents have cargo bikes for transporting little kids e.g. https://www.babboe.de/lastenraeder/big).

vienna is definitely not a nightmare with kids and no car. even disregarding the subway and public transport, i see plenty of people riding around with bike trailers (fits two) if the weather permits.

How about you raise your kids without the big trust-no-one bubble mover?