The problem with this is- if the pc survived that kick to the chest long enough to fight on, why won't the cleric's healing spells keep him alive?
Does basic healing magic in your campaign world regrow lost limbs? I always had the impression that cure light wounds wouldn't give you back your apendages. Then why would it cure a ruptured splean?
If you're in a game world where magic is prevalent enough to cure all wounds, then you are also playing in a game world where death is trivialized by ressurection magic too. In that case a silly death is not really something to worry about anyway right? Perhaps wounds that are grievous enough can't be cured by common magic, which would explain why nobles don't just ressurect their villagers to ensure they can keep collecting taxes.
The bottom line is that you look at a player and say, "You're character is going to die here. Fate has decreed it so. Do you want to go out with a bang or a whimper?" In some game worlds this means roll a new character (WFRP) and in others (DnD) it means getting raised later if the party can afford it.
Do you also give dead bad guys the chance to "fight on"?
Well we are talking "weak deaths" here which bad guys rarely suffer. A paladin commanding his mount to kick the bad guy is not a weak death. It's a death in combat with your enemy.
However, should one of my bad guys die by misfortune, why not. You see it in movies all the time, the guy jumps up from behind the sofa for one last shot at the heroes before being thoroughly put down. It's definitely not something I would abuse, but the same goes for player last stands as well.
You can handwave these issues, sure- but there is no in-game reason why a cleric can heal one dying pc and not another. The burning building guy? A few cure spells will fix up all that fire damage, right? Right???
Just like there's no in game reason NPCs have to die permanently and PCs don't, but it happens all the time. Sometimes magic just won't help, it's why death is a very real possiblity for 99% of a game world's inhabitants.
Plus, even in high magic game worlds, a deity will only let magic do so much. Perhaps the lawful god of death requires that a sure death remain certain once in a while. Ressurection can't be 100% accurate or it risks trivializing one gods power (life) and weakening another's (death).
Although I admire the intent here, my one experience with trying something like this immediately underlined the problems with it. I strongly suspect I'll never do it again.
There are no problems if you a) don't abuse it, and b) decide as a group how foolproof you want magic to be in your game world.