I would have no problem with monsters having death saves. Or even humanoids having healers of their own. But very few NPC or enemies seem to have these kinds of powers. So we either assume that 1) NPC's and monsters are representative of the average creatures in the setting, and the player characters are exceptional. This falls in line with PC tactics, who don't often make sure something is dead while other enemies remain threats.
Or 2) the game rules are not indicative of anything at all, and it's all just an abstraction. The reason monsters pursue different tactics than the players is just how the game is played, and the narrative should not reflect this.
I'm not saying either way is wrong. Or that some other possibility can work for others. I just feel that a DM who doesn't have monsters consistently deliver the coup de grace isn't doing anything wrong or softballing the monsters for his player's benefit- he is trying to have enemies use logical tactics. Or illogical ones, depending on the monster in question.
For my immersion, however, I do try to imagine that all the characters, player and otherwise, occupy the same world, and act accordingly. So if the enemy proves to have a healer, then yes, you could finish off fallen foes (provided you don't have someone actively attacking you who is very much alive). Or you should just prioritize killing the medic.
EDIT: ECMO3, yes, but again there is a degree. Knowing spellcasters exist and knowing what their exact capabilities are, are two different things. But from the perspective of intelligent humanoids, the first order of business may very well likely be to kill the guy brandishing a holy symbol or waving around an arcane focus.
