Why shouldn't they though? It doesn't reduce their NHS work, and it can actually reduce the NHS lists.
What you don't want is private doctors making a botch up of something, and then MAKING work for the NHS doctors to sort out, because that could well happen. Or, alternatively, the best doctors all going private (as many dentists have done), so that the NHS doctors are not so good.
We were at an NHS dentists - the only dentists they could find to work for them were dentists who were working on improving their English before going private. This had 2 results - zero possibility of communication between patient and dentist, which certainly made me anxious about exactly what they were doing, and a high turnover of dentists, so no continuity.
You really don't want NHS doctors to be less competent!!!! And that really would highlight the 'you get better treatment if you pay for it', atm you get better 'lodging's (for want of a better word), but the treatment is good either way (and for acute medicine I would always rate NHS over private), and that's how it should stay.