It is an argument I do not buy, sorry.
If the NHS will not provide safe and adequate treatment for your child - it is perfectly acceptable to say 'safe and adequate treatment should be available for all, and I will fight to ensure this is the case ... but in the meantime, my child needs the treatment NOW, so I will have to, unfortunately, have to ge the treatment privately.
Same for education. DD1 is totally against private education in principle. Both our DDs went to the local primary, local middle and then 6th Form - and we were often criticised to it by most colleagues and relatives who sent their privately. But ... after a few years in their local primary school, it became apparent that juge class sizes and children sent to the school with severe problems without adequate support- GCs education and welfare (bullying and severe disruption of classes, etc) were really compromised. In the end- they chose to go private, with great success. DD1 and SIL really regret that was necessary- and do believe excellent education should be for all.