I agree that the situation now is different from that in 1948, but governments have had 75 years to get to grips with that, and adapt either the funding or the model. It hasn't crept up on us.
I also agree that there are people who expect to see doctors for things that could be dealt with at home, but patients don't aways know the difference between self-limiting conditions and ones that will turn dangerous if left untreated. that's why we need doctors.
We do need to pay more in taxes. I would be much happier to do that (particularly if it could be ring-fenced) than to have means-tested charges for appointments or prescriptions. Perhaps a card scheme could be introduced so that everyone is credited with payments out of tax (or NI or whatever a new ring-fenced charge is called - possibly something new), or as part of pension or benefits (with free credits for children), and you show it when you need an appointment or prescription.
As now, there would be no limit on the cost of treatment, but the card system could register things like missed appointments, and a fine could be applied. It might also cut down on tax fraud, as people working in the 'black economy' wouldn't get the credits that gave them access to treatment, although that would bring problems for people whose employers insist on cash in hand payments. It could open up discussions on things like that though - maybe pub landlords and people employing cleaners should be compelled to pay the 'stamp' of those working for them? Many people live outside of the 'system', which works against them when it comes to things like sick pay and other rights, but they still cost money when they need to be treated.
I think there should be an open debate about what could be done, with nothing off the table, but that the basic principle that essential treatment is free at the point of use should remain. We will maybe need to reconsider what is 'essential' and what is life enhancing, which will mean difficult conversations about things like IVF and cosmetic procedures.