Honestly, 2 strikes down the death save tracker is almost certainly going to put the player on edge. It means that his next death save could be it. His margin of time and ability to recover from a single bad roll has been taken away.
I definitely agree, but when the NPC does that, the NPC is toast, and he probably knows it. They are definitely 100% going to die, probably well inside the next round, because the other PCs are going to absolutely hit them with everything they've got. And the PC in question will probably get hit with Healing Word, so will be back up, on 0 Death Saves (because they auto-reset to 0 when you get up, so even if you get downed immediately again, you're back on 0 saves failed)
This is what I think on GitP they refer to as the "Yoyo", which is very much part of 5E RAW.
Really the only way the players manage to get the PC killed is either:
A) He has 1 death save failed from his current 0 HP status.
or
B) The players manage to do this at the worst possible initiative order, so that a PC causes the NPC to react by the PC's actions, and those actions aren't casting Healing Word or similar, aaaaaaaaand the NPC is also the next person on the initiative order. Given how readying works this would likely mean several PCs has skipped their go or done nothing to annoy the NPC.
In fact, casting
Healing Word breaks this entire scenario unless the PC is a death save down!!!
Here's what happens:
0) NPC has PC on 0 HP held, and readies an action to make an attack if the other PCs take any actions or move towards him.
1) The Cleric says "I cast Healing Word..."
2) The NPC gets his readied attack as a Reaction to that, so before it goes off, stabs the PC, who auto-fails 2 death saves. OH NO!
3) The PC gets healed for 1d6+WIS (or whatever) and is back on say, 7 HP, and has ZERO failed death saves.
Even if the NPC clobbers the PC (unlikely because everyone will blow everything to nail him), he's back on 0 HP and 0 failed death saves. And the NPC no longer has a hostage.
Even if the DM says the Healing Word lands before the stab (not a viable interpretation RAW), the NPC stabs the PC down to 0 HP and 0 failed death saves, rather than two failed death saves, so is now in trouble. Kind of the smartest thing for the NPC to do, perversely, is stab the PC immediately, so he's on two failed death saves, then stabilize him (nothing says you can't stabilize enemies!), then threaten him. That's a bit more reliable.
What have we learned here? Healing Word is a very troublesome spell, as literally any spell that can deliver at least 1HP of healing at a distance!