The younger child goes to a kindergarten on the opposite end of Bad Homburg. That's 40 minutes with the bus one direction. No, there was no option of a kindergarten closer to our house. There was no option at all, you just take it where you get place.
The younger child has an illness and we have to bring him to a therapy at least once a week. That's in a hospital in another sattelite city on the opposite end of Frankfurt. Two hours with bus one direction.
The older child has allergy treatment in another hospital in Frankfurt. One and a quater an hour with public transport.
And these are just a few examples.
Without a car my wife would probably have to stay home just to manage children.
To answer your question - children introduce spatial responsibilities we you can't typically efficiently manage without a car.
The original thought was "everyone should be able to get everywhere they want to go easily without owning a car so a personal automobile should be considered a luxury." You're describing a situation where the first criteria was not met so the second, of course, won't follow. So your situation doesn't apply and it doesn't take away from the original idea.
"Everyone should be able to get everywhere they want to go easily without owning a car so a personal automobile should be considered a luxury" is a nice thought, no objections.
So how about we'll talk about "owning a car ... should be considered a luxury" first when "everyone should be able to get everywhere they want to go easily" is implemented? Because it's pretty far from my reality.
It also looks like Ober-Erlenbach and Nieder-Erlenbach are the only two villages and twos around Frankfurt that don't have a train station to Frankfurt.
Almost every reason I've ever seen given for needing a car have to do with deciding to live somewhere with terrible public transport or lack of public infrastructure nearby.
The argument that there are currently great reasons for automotive mobility isn't affected by the notion that a set of imagined circumstances could enable people to stop being reliant on cars in the future.