Many things we do are 'self-serving', vulpine. Life is hard, there is little safety net, and people/families are often tacking multiple issues, like MH issues, disability, low income etc.
If someone chooses to make their life a little easier by having a car, then I don't think it's for us to judge, unless they are using it frivolously. I try to minimise my own car use, but there are times I think it's justified.
And I agree with others that it is no worse to have a car in London than anywhere else.
I also think it is commendable that the poster (was it pink?) who is running her children to music lessons is doing so. Music lessons are life-enhancing for her children. She is being a good parent by providing them. I live in a neighbouring area to her, and can well believe she needs to travel to see tutors. Everyone I know locally who uses music tutors has to take their children to the tutor, not the other way around (though other tutors, e.g. for maths, do travel to the home of the client quite often).
A trip from Chelsea to Balham would be an absolute ball-ache on public transport, just because of the pattern of the road network and bridges, and the way the bus routes have been set up. Via tube would involve a long trip going around the houses, as the tube lines are largely set up to radiate out from the centre, and do not very well support journeys across the suburbs from east to west or vice versa. That journey would be unviable on a weekly basis.
As long as people are using public transport where they can, I don't see the issue with using a car where justified. It is up to the government to legislate and incentivise greener travel, and provide more of it at lower cost. It has failed to do so. Individuals therefore cannot be blamed for finding a reasonable solution to that problem.