Although the way NICE works in specific instances has led to a lot of very justified criticism, there seems to be no realistic alternative to something like this if you have a tax-financed system that the sick don't pay for directly. There just isn't enough money to do everything possible for every patient.
Who says it has to be tax-financed?
In the USA, much health care is funded by insurance companies that essentially serve the function of averaging medical expenditure among a group of people. The net result of this is that an individual with insurance has every incentive to spend as much as the insurance while let them, because the costs will distribute to all policy holders. When everyone does this, insurance policy costs keep going up. The insurance companies, trying to get costs back down, have incentive only to pay as little as possible, not prioritize approved treatments by QALYs or any other similar metric.
Then, due to all that, you end up with people who are uninsured or otherwise unable to afford health care ending up with emergency conditions, receiving expensive treatment, and being forced into bankruptcy. This also drives up cost (as the hospitals are forced to absorb the cost of treatment) and harms society (a financially ruined citizen accomplishes less and pays less tax).
And thus, we come to this, the worst of all possible worlds.