I'd say this hugely depends on what the player wants to do with his character.
Some players who have invested a lot in a character may want to keep it and not lose it because of a bad roll, or on the contrary be happy to see it die after a long and eventful life or in a heroic fashion.
On such occasion, I (as the player) would have warned the GM either that I had no problem with dying right now, or asked if there was a way I could remain alive (even with some lasting weakness as a result).
On the general matter of life support, I don't like the idea of characters being immortal and knowing that, no matter what, they'll escape any situation unharmed. Players have sometimes to feel a little pang of fear for the life of their characters. On the other hand, I don't like to see a character die while its player had other plan for it, and I prefer in such a case to see a character escape death, but at a high cost. In that particular case, the priest could well have survived with a nasty disability.
"The main difference between Trekkies and Manchester United fans is that Trekkies never trashed a train carriage. So why are the Trekkies the social outcasts?"
Terry Pratchett