Yes, I do think there should be a payment and that we need a social insurance model like those in France, Germany, Austria, etc.
Principally because it might improve funding but also because it would alter the balance of power. At present we have a situation where far too many healthcare professionals think they are doing the subordinate public a favour by providing a "free" service. It is a key factor in allowing sub optimal standards to embed. The NHS is built on gratitude and too many people think they must be grateful for standards that are suboptimal at best or poor at worst.
My GP practice does not do me a "favour". It provides a service for which I pay handsomely via taxation. Too often, the fact it is "free" allows those running it to design the service for their needs rather than for the needs of the patient. As a petson who works full time (as hard as most GPs), I'd be happy to pay a premium for an appointment before 9am or after 6pm, booked in advance.
The bureaucracy of the NHS is absurd. For 35 years I've had free prescriptions across the board for everything, not just the essential Levothyroxine. DH got free prescriptions the minute he turned 60, we both now get free eye tests. It's ridiculous. We are both still working, both earning over £100k and even when we retire will have a combined household income in the top 5%. We don't need free prescriptions or free eye tests. Free access should be for those who need it.
In relation to dentistry, our lovely dentist, who we have always seen privately, withdrew a couple of years ago from NHS work. Not for financial reasons per se but because the treatment allowances are so pitiful they can't provide people the quality services they need and it compromises the professionalism of doing an acceptable job with the time and materials allowed. The final straw, however, was not that but the heavy handed bureaucracy of the commissioning bodies that made the administration of claiming for the work disproportionate, difficult and officious.