I don't see what the issue is here.
The child is entitled to a state school place plus transport if it's over 3 miles away. The council is legally obligated to provide child with.a school place. The only school place was 25 miles away, so the council has to pay for transport.
The parents being able to afford it/able to home educate/whatever makes no difference to the child's legal rights. The child doesn't lose her rights because she previously attended a private school.
It's a bit of a shame that this child is now costing the UK taxpayer £7,000 odd for a school place per year plus £8,000 per year in taxis, so £15,000 that the taxpayer previously didn't have to pay. But that only matters if we think the reasoning behind the VAT policy is financial rather than ideological. If we accept that it's ideological, we can't really object to parents making use of the state provision (including transport costs) to which they and their children are entitled.