Whilst I understand that reasoning, I disagree with it. Private insurance should not be necessary to begin with, and if it must exist for whatever rational reason, the insurance should be public as well. Anything that has to do with healthcare (not talking about optional cosmetic surgery, of course) should be handled by / via public healthcare services.

Furthermore, I would say that it's wrong to look at the situation in terms of "reducing the burden on the NHS Dental", since that is literally what's there for. If people don't opt for cosmetic surgery, then it's a healthcare issue and we created this institution/branch for this very purpose. Otherwise we can easily end up speaking in terms of how much of a burden being alive is, and how eating (regular) food and drinking water are poor economic decisions, a topsy-turvy way of looking at the very reason to have an 'economy'.