I don't see a reason to make being nearly murdered more punitive- it can happen at any time, say due to a lucky critical hit (because unlike most PC crits, a monster can generally toss a ton of dice your way).
You have a great adventure lined up, and you're really excited to see how the players handle the final encounter. You've given them some time sensitive reason to hurry, perhaps an evil wizard is about to do something nefarious to a beloved NPC.
3 encounters into the 5 encounter session, a Giant Ape throws a rock at the Cleric with a natural 20, and down he goes. Thankfully he can be revived with a potion of healing, and the adventure continues, though now the players are down more resources, because they don't want to risk taking a short rest.
But wait! The campaign is using a house rule to make going to 0 more punishing, because "verisimilitude", and the result is that now the Cleric is suffering from some lingering malady, placing the odds of success even further away!
I know some of you are nodding your heads, thinking: "yes, that's how the game should be, adventuring is dangerous". But I would think that making the Cleric use up a bunch of their healing in this scenario, when they really need to conserve resources for the final encounter, is already punishing a player for something they had no control over.
There was no way the Cleric could have avoided this fate by pre-emptively healing himself with cure wounds. He might have given himself 5 temps with Aid, but even if that would have helped, who would have foreseen needing them?
It's basically saying "yeah, you should have been a Twilight Cleric".
And let's not forget the other fun thing about death saves, and why being able to revive characters quickly is important to the system; if you happen to take damage while you're unconscious, that's an automatic failed death save. Saw that happen when a Barbarian went down after being lit on fire.
It isn't just that Healing Word is a bonus action that can bring people back from the brink is the problem either; it's a ranged healing spell, where cure wounds isn't; if someone goes down, the Cleric might not be able to reach them to cast a real spell even if they wanted to!
So perhaps tetrasodium has the right of it; if you want to change the healing dynamic, you can't just throw out a house rule and call it a day. The entire way healing and death work in 5e would have to be rewritten.
Or we could embrace the Twilight Cleric's method of "healing". Which some people find equally as problematic.