But from where I sit, the dying doctors' point of view, as presented in this article, struck me as irrational.
Axiom: Being dead is better than being alive. Under all circumstances.
I personally believe this axiom, very strongly. I'd rather spit in the eye of death with my last breath than go quietly into the night.
It logically follows, then, that a treatment that has a 1% chance of working is better than doing nothing, if the latter has a 0% chance of working, no matter how extreme the treatment is.
Living longer, even in a state of constant extreme pain, is better than dying. Well, maybe dying peacefully would be less painful for my relatives, but again, I personally place a much higher value on my own life than my family's feelings of sadness or whatever.
The only question which remains is the validity of the Axiom. If this world is all there is, if when you die there isn't an afterlife or reincarnation or whatever...then the Axiom is clearly true.
If you have strong religious belief, then perhaps your afterlife is better. But only a perfect belief that you are headed to a better place makes it rational to give up. If you have a little bit of doubt -- if you're 99% sure you're going to Heaven but 1% of you whispers that maybe there's nothing and you'll turn into dust instead -- you still want to put off rolling that die as long as you can. Ditto if you're 100% sure there's an afterlife and 99% sure you've been good enough to get into Heaven, but 1% of you says that maybe Hell is where you're headed.
If your religion forbids suicide, and there's a chance that the Judge who will weigh your life will interpret not opting for the best available medical treatment as equivalent to suicide, then that affects your decision too. Although, to be fair, this cuts both ways: The suffering you cause your family by lingering beyond your time on Earth might count against you too. Literally "Damned if you do, damned if you don't." It's unclear enough that there should really be a few more commandments or something, to address all the corner cases.