According to the Great Sage, the Dark Curse, after somehow removing his face, was unable to cling to existence. Remember all those people whose faces you saved? Let's just say they got very, very lucky...
However, the Dark Curse was removing OTHER people's faces, and they were fine. Removing his OWN face, on the other hand... yeah. So, perhaps it's a case of "removing-your-own-face-is-lethal"?
It's more so implied that the Dark Curse lost his physical body because he was without a face for so long. All the other Miis were probably able to get their faces back soon enough to keep their physical bodies. Had they been faceless for long enough, they too might have died.
When a Kind Mii successfully spares a monster, they run away. Which means there's someone out there who's not getting their face back.
The ending of the game says that the monsters all disappear and everyone gets their faces back when the Dark Curse is no more. Meaning every Mii gets their face back eventually; it’ll just take a bit longer, in certain cases.
It could also be that the monster later removes its own Mii face. You don't get the credit, obviously, and the critter returns to being harmless.
Might also clash with Fridge Funny, but some monsters contain only a mouth, a nose, or even just one eye. Meaning that there's a Mii who's only missing a part of their face.
Alternatively, some Miis just have their missing faces split between several different monsters.
The Dark Sun is made up of plenty of Mii face parts, and while all the other bosses in New Lumos release their stolen faces as normal when they die, he instead explodes quite spectacularly. The counter for your number of rescued Miis doesn't even go up, meaning those faces got possibly destroyed outright.
Also about the Dark Sun, consider for a moment a few factors; the Dark Sun is a more powerful version of the Darkest Lord, the One-Winged Angel form of a character with the combined powers of the Dark Curse and the legendarily strong Great Sage, and a form so powerful that it requires a full team to defeat one of its hands. The Dark Sun can also, unlike most other bosses in the game, be fought as many times as you want, even despite its certain and total destruction, and it has a new face (and new face parts) each time. This is a monster more powerful than what might as well be a god...and there's multiple of them.
The Prince of Neksdor is in the middle of stewing over the princess rejecting him when he encounters the Genie of the Lamp, and is coaxed into freeing him awfully easily through the promise of wish fulfillment...Was he planning on wishing for the princess to take him back, completely against her will? If so, it’s probably a very good thing that the Genie ultimately reneged on his end of the deal.
There's post game dialogue from traveler quests where the traveler speaks of the world they saw through the eyes of a monster, confirming that those with their faces on monsters can see what the monster is doing. The implications are horrifying enough, but now, think about the Greenhorne arc again.
It's downplayed due to them being a minor role and thus more likely to be set to someone silly, but the Sassy Child's behavior becomes significantly more disturbing if you're aware that young children often express their trauma through play...
A Fridge Tearjerker example: if a Mii has the "nightmare" status and isn't in the safe spot, their dialogue for that turn will be, "please stop fighting!" Considering the quarrel mechanic, perhaps they're dreaming about their friend group falling apart?
The King spends most of his time eating. When his face is stolen, he remains in a panicked state until you save him, at which point he says that he needs to restore himself and starts eating again. Could this mean that when a Mii loses their face they lose the ability to eat?
What would happen if the Dark Curse possessed the hero Mii...?
