[vidarh]

Effectively many countries do allow this to some extent.

E.g. a significant portion of private care in the UK is provided in NHS hospitals by NHS staff either outside of their normal shifts arranged by private companies who pay the NHS for access to operating theatres and the like, and by NHS trusts themselves who are allowed to provide private services on top of their socialised care.

A market exception is Norway, which as one of very few countries have very strict limitations on what private services can be provided at all on the basis of an argument that as long as healthcare resources are limited (e.g. there is a practical cap on number of doctors - train too many and they don't get enough experience on real cases to gain the necessary skills), it is seen as immoral to allow people to queue jump.