Morkus from Orkus
Before I was speaking as a player. Now I'll come at it from the DM side of things.Because the player is never objective.
No matter what the player decides, it will always come from the very subjective position of the player. Even players who delight in torturing their characters are still making choices based on that particular preference of play. IOW, the player will nearly always choose a response that the player thinks of. Obviously.
Take a DM's example. The monsters have downed a PC but there are still standing PC's around. The monster's turn comes up. Now, as the DM, you could instantly kill the downed PC - two automatic death fails kills the PC in this example (assume the character has already failed one death save). So, as the DM, do you whack the PC or attack someone else? Well, either way you decide is tainted by your awareness of the table. If you choose to kill Dave's character, he might be kinda pissed off. OTOH, if you choose not to kill Dave's character, are you making that choice because it makes sense in the fiction or because you just don't want to kill Dave's character? But, if you kill Dave's character, are you doing it to avoid looking like you are avoiding not killing a character - on and on and on, around in circles.
So, if you're me, you let the dice decide. 1-2, kill Dave's character, 3-6 move on to the next target. It's objective and fair and doesn't put me, as DM, square in the spotlight for whacking Dave's character.
The same goes for players. Players will never choose something that they don't think of themselves. They can't. Obviously. So, that's where mechanics come in.
As a DM I much prefer to have control over whether the monsters kill the downed PC or not based on the circumstances surrounding the fight. Below I'll give some examples.
Example 1: The PCs are facing ravenous ghouls and the downed PC has a ghouls standing over him. Ghouls aren't that smart and really exist to eat human flesh. The ghoul would stop and start eating the PC, so the PC gets attacked while he is down. (this happened in one of my campaigns and the druid died).
Example 2: The PCs are facing an experience mercenary band who knows that downed enemies often don't stay down, so tactically the best thing to do is make sure they don't get back up. The mercenary over the downed PCs finishes the job.
Example 3: The PCs are facing off against townsfolk who have been enraged by an anger entity that is plaguing the town. The downed PC has farmer Joe standing over him. Angry or not, farmer Joe doesn't know tactics and isn't familiar with enemies getting back up in battle, so he just moves on to the standing PCs and tries to attack one.
Example 4: A new town guard that has only had a few weeks of training is standing over the PC. He has some small training in tactics and the idea that enemies can get back up, but he's very new and there's a good chance he's not going to think of it in the moment. I don't know for certain which way it will go, so I'm going to assign 1-2 he kills the PC, 3-6 he moves on to attack the others.
I don't want there to be set general mechanics for making that decision, because those mechanics cannot take all the details of the moment into consideration and give me what I consider to be reliable results on what the monster/NPC would do.