As an English person living outside of the U.K. it is always interesting to me how much more British people who have never lived outside of the U.K. rate the NHS compared to those of us who have lived under other healthcare systems.
That isn't to say it is bad... it has some amazing points - most obviously and notably the fact that it is free. But when I'm back in the U.K. it often feels like it is treated as a theoretically perfect healthcare system that is only let down by the pesky government not giving it enough money...
My worry is that this over-confidence ends up obscuring the fact that it does have some significant problem (and advantages) when compared to the huge variety of other systems that need to be addressed.
I would absolutely say that one of the biggest issues the NHS has in comparison to other partially paid systems is the wait times. I'm always shocked when talking to my U.K. family and they tell me about appointments that lead to booking "tests" and procedures which they then have to wait weeks or months for. Where I am, we would have to pay a small amount for each test, but the tests are often same-day, or at works same-week.
Surely there is a middle ground? Where those who need it can get it free, and those who can pay, pay. I know this offends people on principle, but in practice from what I've seen outside of the U.K. it can raise standards for everyone.
Does it feel uncomfortable that some people pay more for better flights? Yes. But in doing so their fees subsidize the entire flight and enable such plentiful and cheap options that we all use.
If, hypothetically your choice is:
No paying, everyone waits 4 weeks...
Or
Rich pay = done tomorrow
The money is fueled into lowering free wait times
Non-paying = done in 2 weeks.
Yes you have created a 2 tier system, but as in most industries, the 2 tier system results in the lower tier actually getting a better deal than if it is all entirely equal.
(and this isn't even a good example, because we already have a 2 tier system, just at the moment all of the profits go into the hands of private medical companies rather than back into the NHS where they could be used to raise the free standard)
So yes, I think you are roughly right OP. But you won't be popular for it in the UK.