One of the underlying problems is that we haven't really faced how to deal with care as people get older, and we also have really expanded what counts as healthcare a lot since socialized care came on the scene.
Back when the NHS was developed it was seen much differently in terms of what should be covered. Not just explicitly, but also people's underlying sense of when they should look for care, which is a really significant element of how the service is used.
I think part of MyPithy's point is that we know all of us will eventually die, no matter what interventions we make. There is a decreasing cost benefit to them, which is an important thing to consider - as much as it seems off to us, cost is an important part of how we decide what health care to fund.
For example we might fund one childhood vaccine, because it pays for itself, but not another because the cost doesn't justify the benefit to the system - and the limited money in the system would be more effectively used for some other initiative. That seems to be the part people forget when they are talking about a system with limited resources - yes, you can choose to fund very inefficient treatments "just to save one life" but the fact is you are then not funding something else.
But with care as people become elderly, it's not just the moetary element, there are also questions about quality of life, and quality of life when we extend it significantly through interventions. There have always been some people who live very long lives in fairly good health - but the important point is those are people who did not require many serious interventions. They were not having heart surgery, or living with heart failure for decades into their late 80s, or cancer treatment. They lived a long time because they were naturally lucky and had a long term healthy lifestyle.
When people have very interventionist treatments in old age they may live longer but often with a very poor quality of life.
Does this mean they should be refused treatment - no, not necessarily. But I think it does suggest we need to rethink our attitudes to treatment in various ways, and maybe how they are presented to patients and system users. Medical people are often much less likely to have these kinds of interventions themselves, which should tell us something. Part of the problem is people don't face up to the fact that death, and the things that lead up to death, are inevitable - and they will face them eventually whatever treatments they take now. Making choices because we fear death itself is fundamentally illogical (though very human) because nothing changes that particular outcome.