I know quite a lot of families who live in London and other cities with ULEZs without a car. If you chose your work, school/nursery and home locations based on what you can reach without a car then it's fine. Public transport isn't the drama people make it out to be (babywearing and quick-fold pushchairs that can be collapsed with one hand). There's a lot more use of cargo-carry type bicycles to get younger kids around. In one such family their eldest has just finished their first year of senior school and their daily commute has been a 15 minute cycle ride to an overground station followed by a 15 minute train ride (trains every 15 minutes) and a 10 minute walk, which has been totally fine to do solo. The children only go in a car about half a dozen times a year when staying with family for holidays.
Another family with younger children have a cargo bike kitted out to carry the kids. On days when both parents have to be at work, in the morning parent A will take the kids in that to school/nursery and leave that bike there and take a borisbike to work and then come home later directly by public transport, meanwhile parent B having gone to work earlier in the morning by public transport will return via nursery/school and pick up the cargo bike and kids. They are members of a car club and can use a (ULEZ compliant) car that is parked within a few minutes walk from home if needed, and get preferential rates on hire-cars for holidays.
I live in a non-london ULEZ and I have a compliant car which I only use a couple of times a week. It certainly makes life easier to have it but I could do without it. In principle I try to only use it when the journey couldn't be done by bike/on foot/public transport but about half the journeys I do are in reality things I could have managed without a car if I had been more organised/less scatterbrained but those traits are part of my ASD and aren't going to go away, so I just do my best. DC schools & nursery have always been either in walking distance or reachable by public transport, me & DP have jobs where we can either get there on public transport or by bike when we don't work from home.
There are certainly vast swathes of our city which are built for the culture of car worship. I would like a house that's a bit larger so am keeping my eyes open on rightmove to see what sort of thing I could aspire to. There are entire regions of the city outsulide the ULEZ that I just couldn't consider because the public transport options are inadequate. But the ULEZ is part of fixing that - the LA have got to start having an integrated policy that reduces car-reliance by making other options easier, and this is a cultural shift. If houses in locations that can't be managed without a car become less popular and drop in value because you know it's going to cost thousands more every year to live there, then the house building companies reassess the cost/benefit balance between building on greenfield sites at the edge of the city vs brownfield former-light-industry zones near the centre which may be more difficult to build on but have loads of public transport options.