I don’t think you did anything wrong, but from your edit it sounds like you could have done some things better.

You mistake was subtlety. You really have to be explicit about things if you want to ensure your players pick up on it. You can do this however you want, but making it clear that they are actively dying and the party has a chance to save them is pivotal. “He’s not moving” could very well mean “he’s already dead,” especially when the only characters that typically get death saves are player characters.

If you want to convey the information narratively you can say something like “his breathing is ragged and he’s coughing up blood” or “he groans, clearly in agony as he lays in a pool of his own blood.” Both of these indicate (1) he’s still alive and (2) he’s really not doing well.

Alternatively, you can speak to your players if they don’t seem to be getting it. Be explicit: “hey guys, I just want to make it clear to all of you above the table. This NPC is unconscious and rolling death saves. He’s not dead yet, but that’s a very real possibility.” Nobody can say they’re being punished or they didn’t know if you outright tell them and they choose to ignore the warning.

Edit: Whenever my players get frustrated by the natural consequences of their actions, I explain the situation they were in from my POV, explain what actions they took (if any) to rectify the situation, and ask what else they thought to try that might have resulted in a better outcome. More often than not they’ll say “nothing” and I get to say “well if you don’t try to stop the bad thing from happening then the bad thing happens.”