I think there is a distinction here in what sort of deaths they are intended to protect against. Much like 3e's negative hitpoints and bleeding out rules, they serve as a means for the player to recover after a bad luck streak. I don't think that is a bad concept. In fact, I think it is very important. A lot of 5e's combat is balanced around this rule.
Where it fails, is in the ease by which hp can be restored. If this problem didn't exist, death saves would be fine as is.
What also doesn't help, is that 5E has removed most of the save or die effects from monsters. For example, the Medusa's petrifying gaze is a lot more forgiving than its 3e counter part. And the demilich merely drops a player to 0 hp, instead of instantly trapping their soul and reducing their body to dust.
Surely we can keep 'a safety net' in the form of death saves, while also making combat more deadly? We could limit ranged healing for example. Or we could limit the healing of incapacitated players. We could apply penalties to having been incapacitated, that linger even after stabilizing. We could re-add the deadly abilities that notorious D&D monsters used to have (although it might be confusing to have multiple versions of a monster). There are all manner of things we could do, that don't remove any of the core rules.
How about a condition called At Death's Door?
At Death's Door
Any player that recovers from 0 hp, can no longer make death saves for the rest of the day.
Or a more extreme version:
At Death's Door V2
Any player that recovers from 0 hp, can no longer make death saves for the rest of the day, and can receive only half healing, and makes all their attacks and saves at disadvantage.