I think it's also possible that the "withholding of death" is not permanent in these cases, but rather a temporary delay. I think of these ringbearers as not completely immortal, but rather infinitely stretched - part of their fea wanting to seek the gift, and the other part being held in place by some attachment, thereby stretching it (like butter spread too thin, as one of these lovely people described it
). One main aspect of this is that while death/biological processes are delayed, they are not stopped. It's almost like one life is stretched over the time of many, so that ageing slows down, but also the "intensity" of life weakens. Like you can only live so much, have only such a concentrated fea, that when life is stretched it can't be saturated across all it's length. Hurin's case is notably different from the others, but I want to point out right away that he went from Thangorodrim as an old man. Morgoth may have forced onto him a supernatural strength to survive without basic needs, but this was a physiological kind of interaction. Morgoth just kept his body in sufficient condition to hold his fea (which was still young enough not to die by itself... you know what I mean!); he did not interfere with the spirit. I don't know what would have happened had he not lifted his curse - and I'm not a great fan of the realm of the would-haves; I'll just stick to what there is. The ringbearers, on the other hand, are described more like their entire lives and fear are stretched, not just some survivalist ability. I think that's part of why tge ringwraiths aren't really alive - because part of their fear are already dead, and only the very remainder that is held by the hook holds on. Once the hook is removed, they would die. So it's like a very long but temporary delay of death.
Another thing is that if a Ring was given to a dying person, I don't think it would prevent their death - like Mithadan says. Once the Gift is given... well, only Beren came back, and even then only death can pay for life, as GOT fans would say. Instead, the Rings seem to stretch the middle part of life, such that the time for the gift does not arrive for a possibly infinite amount of time. And there is all the stretching again, where what should have been a more concentrated 1 year of life becomes a significantly more dilute larger number.
Hopefully I'm making sense with this. I'm trying to keep my thoughts apart, but the points are so intertwined that it's hard to keep them as separate ideas.