I'm familiar with the area and schools. It's true that the closest schools are all full, however the mother could have appealed for any of them, and would have had a good chance of being successful as all schools were full.
I suspect when she contacted the schools within a 20 mile radius of her house, she only actually spoke to the schools she preferred and not actually ALL schools.
She could also have used the fair access protocol to be given a place as again all schools close by are full.
She chose to apply to a school a ridiculous distance away as it is allined with her aspiration lifestyle, she chose to accept that place and then drove the child for a period of time before applying for transport, it's quite likely she did this to try and establish the child in that school to then use that as leverage to gain transport.
There are places available at other schools closer to her home, but they are not "good" schools so she doesn't want her child to go to them.
It's not true that this particular, high performing, school was her ONLY option, if she wants her child to go there and the school are willing to admit the child than that's fair game. However given that it's a parental choice she should either drive the child herself or use the money she's saving by not paying school fees to fund the taxi herself. If neither of those options are acceptable to her then she should have accepted that her preferred school isn't a viable option and done what everyone else has to do and accepted a less preferred school that is serviced by the (must cheaper) school bus service.