Ho boy, that kind of thread again.
Whack a mole is particular to 5ed. I have stopped the WAM exactly by doing the OP way of doing it.
Some will call it meta gaming and I do not.
1) Foes will not terminate a character unless one has got up first. In essence, if a character falls and only a spare the dying was cast, then foes will not try to "finish" the PC. But if the character gets back up with a healing word or better; it will be a signal for the foes to start making sure that the enemi will not come back to haunt them.
2) Foes will behave differently if they are intelligent or not. An undead might try to get a quick snack or an animal might try to get away with its prey. But an intelligent foe will act upon both what the players will do and the general knowledge they might have. If the players know that fire is needed to end the treath of a troll, so should their enemies know that adventurers can comeback from fallen foes to killing machines in an instant.
3) Once a downed PC gets back up. Intelligent foes will target the cleric or the healer in priority. Even then, if too much healing is done, a healer is better prepared to get a defensive spell or two at the ready. Foes will try to kill him ASAP.
And for those who argue that this is meta gaming. It is not. What players can do, so can their foes. Yes it means that the lethality of my games is high, but this is the goal I had wished to obtain along with my players. I often see some of them standing up during tense fights and they do cheer when they win a particularly hard fight. This forces my players to make and use better tactics and think of imaginative and safe ways to defeat their opponents. So far, I have had no complains from any of my players and pre pandemic, those that were coming to watch our games at our Friday night dungeons always found this gameplay to be both challenging and interesting. Not all fights are to the death, monsters flee and so are the players when the odds are against them.