Overall I'd agree, but with a few points:I think that the main reason that the "down-and-making-death-saves" system was added to the game wasn't to remove fear, but to add it, as well as increase player engagement and increase tactical complexity.
Options to "just don't take damage" are limited in 5e, particularly for the types of character that tend to be on the front line defending the rest of the group, and healing couldn't keep up to mitigate it either.
A character dropping to zero and dying immediately loses any engagement for their player, except by proxy for their friends for the rest of the fight. For the rest of the party, there is no response or counterplay when the DM rolled lucky: the character is dead and that's it.
No pressure, no build up of tension, just the wait until the new character can be introduced.
A character going down and starting to bleed to death immediately adds tension and changes the tactics the players need to use. There is now a new and probably overriding objective in the fight: protect and revive that character. Can the fighter get there and push the monsters away from their comrade? Should the Rogue deliberately provoke an attack of opportunity on their turn, so that the druid can safely break off to go heal the downed character? Does anyone have any potions? It isn't just the player of the downed character feeling fear, the entire rest of the party feels fear for their endangered member and have to shift gears to try to save them.
Your reply sort of demonstrates what I'm saying: you're talking about near-death drama because in 5e, a character hits 0 and is "dying."
But if you have 30 max hp, but you have 4 or 5hp right now and you die at zero... this is basically your last stand, looking death in the face. There's tension, drama, tactical decisions to be made, all right there. Because right now, at 5hp, you're probably one blade in the ribs from death. But as it is now, with falling unconscious and dying, we just don't look at it that way.. because you can get right back up with a touch of healing and be hunky-dory.
I don't disagree that having a different state, "unconscious and dying," couldn't add more to it.. but 5e immediately plasters over that with stuff like Healing Word- a 1st level bonus action 60ft ranged spell that'll get them right back up.

