I stick to the RAW. I have no issues with the death save mechanic. I have no issues killing players at low levels. I'm a big fan of the funnel system where players start with four level-0 PCs. From any that survive the first adventure, they chose the once to make a 1st-level character with. The idea is that everyone is going to have at least one of the four characters die and their is a chance that all four will die. It helps players get used to the idea of characters dying and teaches them to be careful and bit more tactical early on.
As for whether I have monsters continue to attack characters at zero hit points, it depends on the nature of the creature and the specifics of the encounter. Many times in the heat of battle, you see someone go down and you move on to the next immediate threat. But there are plenty of times when this won't be the case.
A roc, who is just looking for a bite to eat, may just attack until a character is down, and then carry him off. Same with other carnivorous flying creatures who are able to carry away a medium-sized creature. Also, in a world with gian owls and eagles, why not giant vultures? They may circle battlefields and when they see someone go down they swoop down and carry the 'carrion' away. If the character makes three death saves and comes conscious, the vulture can just drop the character from a high height to finish the job.
If not in melee range with other characters, an enemy who takes down a character may make a follow-up attack to stab into the heart or cut of the head, similar to double-tap drills taught to many shooters today. I don't mean walking around stabbing corpses after a battle to make sure they are dead. I mean, a soldier could very well be taught to make a final attack after taking someone down--if combat situation allows. This would be especially the case in a world with magical healing I would think.
Then you have those who are just in a rage or blood frenzy. They don't want to just kill you, they want to rip you apart.
Other situations where enemies won't stop at zero:
Zombies, after you go down they start to devour you.
Swarms. Why would a swarm of stirges, or rats, etc. tactically move on to the next party member. They would continue to feed on the fallen. The party will have to fight the swarm off -- at the risk of further damaging the prone and unconscious party member. Swarms should be a scarier than they are typically played.
Oozes, jellies, and puddings.