I find it so weird that people constantly speak as though public transport is this hypothetical maybe like a moon base or something. I use exclusively public transport, bikes, and walking. My whole family (with children) does. It's just not a problem.
Children walk or cycle to and from school. By themselves. When they're very young their parents did go and pick them up sure, but then its a small school within walking distance.
We rented a car and a trailer for a couple hours recently to move a double bed. It posed no problem, and was dramatically cheaper than owning a car for a month would be even if the car itself was free.
I got a nice cabinet for my friend recently. We are going to take the drawers out and move it to his place on a bus. I don't think it would fit in your car.
Replacing most of car traffic in a city with public transport (and bikes and such) is possible, and it can work - after it stabilizes. The transition seems extremely disruptive, which might be why people speak of it as if it couldn't work.
I'm a parent with small children. We have a car, but we only use it for inter-city commute. Everything within city bounds, we handle by public transit or walking (or electric scooters). It works because we live close to the kindergarten, and close to multiple public transit hubs, and I work remotely. It works, because we planned for it in advance.
Now, take a typical car-commuting, office-working parent of today. Like most, the place they live in is frozen in some balance between their and their spouses' jobs. Changing it would upend someone's schedule, and possibly involve kids changing schools/neighbourhoods (which isn't good for them). At that stage in life, one's combination of home, workplace, kids schools and after-school activities, is pretty much frozen in place. If they made it work with car commute, it probably can't work with public transit, and thus if you suggest the change, they'll look at you as if you came from Mars or something.