The difference is in the NHS everyone is "on the same side", so the amount of bill inflation and money juggling that happens is basically zero.
The NHS does demand management with waiting lists instead. Basically it's amazing for things that can be easily identified and given cheap medication for (insulin, antibiotics etc), amazing for emergencies which can be resolved with surgery, OK (but variable) for obstetrics, does a decent job at screening for common conditions, but tends to leave anything that won't actually kill you to wait.
At the same time you can get super-cheap private insurance in the UK because they are effectively a "queue-jumping service" that leaves most urgent and lethal conditions to the NHS and gives you access to private treatments for the things the NHS would put on a waiting list, while cutting their costs by renting excess capacity from the NHS (some private clinics are even run by NHS trusts, as the trusts are allowed to derive some income from private treatments).
I think the best demonstration of the quality of the NHS is that despite the low cost of private insurance in the UK, and despite the fact many companies offer health insurance as a perk, only about 10% of the population has any kind of private cover.
The NHS does demand management with waiting lists instead. Basically it's amazing for things that can be easily identified and given cheap medication for (insulin, antibiotics etc), amazing for emergencies which can be resolved with surgery, OK (but variable) for obstetrics, does a decent job at screening for common conditions, but tends to leave anything that won't actually kill you to wait.
I've never had to think about billing.