The OP points to the German health system as an alternative. I'm not sure why (other than ulterior motives or lack of understanding) the only alternatives discussed are the NHS as it is - or fully private provision US-style.
True story. I'm in France. I have hurt my back, and I've been feeling really run-down and tired, so I make an appointment with my GP. I do this online, and get an appointment in a couple of days time. I go to the appt, the Dr examines me and gives me three prescriptions - one for anti-inflammatories, one for physio and one for blood tests. I hand over my Carte Vitale (national health card) which links me to the government health system and I pay 25 euros for the appointment.
I drop into the pharmacy on my way home. I hand over my prescription, my Carte Vitale and my mutuelle (top-up insurance) card and pay for the medication.
I also drop into my local Laboratoire and show them my prescription. They say that my blood tests need to be done first thing in the morning, without eating or drinking, so I can't do it there and then - but I can come back any morning that week btw 0730am and 0930am - no appointment needed.
Once I'm home I go online and look for physios near to my house. I find one that's close by, and I make an appointment that fits round my other commitments. If that one doesn't have any available appts then I look at the others around me until I find one that suits. I could also ask the Dr or friends for a recommendation and then make a rdv with them - but I like the convenience of booking online.
The next morning, I skip breakfast and head to the Laboratoire before work. I'm met at the door, my details are taken, and I am taken to an individual treatment room where the nurse takes my bloods. When I'm done, she tells me that once I've paid, I can come back the next day to collect my results. I pay - again having handed over my Carte Vitale and mutuelle card - and leave. The next day, I drop in and collect my results - and make another GP appointment to discuss them.
I go to the first physio appointment. Like most physios, she has her own office, treatment room and gym - all private, just for her. Our weekly sessions will last for half an hour each time, and includes massage, exercise and other treatments. The sessions come in blocks of 8. If I need more after that, no problem: I can extend this.
Again, I pay the physio up-front for the sessions once a month.
For all of the above - GP appt, medications at the pharmacy, physio sessions - I am reimbursed in full. The government (through my Carte Vitale) will reimburse up to a set amount, and my mutuelle will reimburse the rest - all direct to my bank account. Yes I need a certain amount of money to pay up front and yes I pay a monthly premium for the mutuelle (and it's a lot less than private health insurance), but by and large I am willing to pay that for the level of healthcare provided. My mum has been to an NHS phsyio - waited months for an appointment, then the "treatment" consisted of 1 session and a badly photocopied sheet of exercises to do at home.
It works this way for virtually everything that our family has needed over the years - pregnancy and birth, broken wrists, speech therapy, dentistry, x-rays, blood tests, etc. There are some treatments that aren't fully reimbursed, and the up-front charges depend where you go (public hospital v's private clinic), but by and large I am really happy with the healthcare here.
It's a half-way house btw private and state care in that the providers are more or less private - laboratory chains, pharmacie chains, individual care providers like phyios and nurses etc - but the cost of using them is covered by the state. So the state doesn't have to get so involved in the actual provision of care.
There are provisions in place for those on very low incomes, they can apply for a specific Carte Vitale.