The problem with British people's opinions of the NHS is that they routinely conflate the NHS with healthcare. (BTW, I am British myself). You're doing the same thing here.
The NHS is not healthcare. The NHS is not doctors or nurses or life saving operations. The NHS is not hospitals. All first world countries have these things, but none have an NHS except Britain.
The NHS is a bureaucracy that manages resource allocation. That's it. That's all it boils down to.
When you saw life saving operations working as hoped, that isn't happening because of the NHS. You'd see the same in France, Germany, Sweden, the USA, Japan or many other places. We don't judge the skill of a bureaucracy or institution based on the skill of individual employees, we judge it based on factors like:
1. How much overhead does it impose?
2. How competent is it at organising its operations?
3. How agile is it, how able is it to react to new circumstances and continually improve itself?
4. How many people can access its services, who might want to use them?
5. If it fails at any of those criteria, how easy is it to switch to a competitor?
The NHS varies from average to poor when evaluated by these criteria, with the exception of (4) where it gets the best possible score because it even happily treats people who flew in to the UK specifically for free treatment! But if we exclude that edge case then it becomes pretty average again, because universal access is solved in other societies using insurance schemes of various kinds (sometimes mandatory and subsidised). The exception is the USA where for cultural reasons a lot of people don't like being forced to buy health insurance.
Although its overhead is reasonable when evaluated in pure GDP terms, this is achieved partly through building up large maintenance backlogs which is hardly sustainable: true cost of the NHS to the UK should probably be higher than is actually reported. The government has tried several times to force the NHS to spend money on maintenance and upgrades but usually fails: the NHS takes whatever money was granted to it for this purpose and immediately spends it on daily operations in violation of their agreements. Nothing happens because to Brits the NHS is a holy religion, so NHS management don't really feel accountable.
The biggest problem with government run industries is not that they can't carry out their mission at all. Soviet factories successfully made steel and rockets, after all. Their problem is lack of agility and lack of quality. Agility: see the NHS testing ramp-up problems discussed in sibling threads. Quality: see how slowly the NHS ramped treatment back up after the April shutdown. Other countries did much better in that regard, because private hospitals desperately want to treat people in ways that the NHS just doesn't. How could it be otherwise: if private hospitals don't treat people then they run out of money and go bust, like any other business. If the NHS doesn't treat people, its staff basically get a paid holiday with no negative consequences. Of course that affects people's behaviour.
I said average to poor, that's true except for (5) where the impact is catastrophic. Its tax funded status means the private health sector is seriously throttled in the UK. So only the rich can work around NHS failures, and even then not always (private hospitals don't do the full range of treatments in the UK).