You're making a lot of assumptions here. They don't prevent the thing that woke them up because the thing that woke them up was being attacked, which they were. The resolution of the attack is what the combat rules are for.I mean sure, it's my opinion, but I'm having a hard time figuring out what you're doing here.
Assuming the character is quiet while sliding a blade into someone, they aren't waking up until the attack has already hit and damaged them. It's the damage from the attack that causes them to wake up. If you run it so that they wake up before the attack and damage has resolved, they are in a whole different boat, and can theoretically prevent damage from happening (assuming surprise isn't involved and preventing that, which I think we agree normally is), which would mean that they wake up and prevent the very thing that woke them up!
That's what I'd be having a problem with. If the player starts combat by bellowing a war cry--yes, I'd have the foes all wake up. If they started combat by charging in full clanking plate, I might wake up the foes. And maybe that's what's happening. But the image I have is that they have their sleeping foes surrounded, ready to kill them in their sleep, and right as the first person quietly stabs/slashes, the creatures all wake up.
As far as initiative, I'm not sure I run it much differently than you described it, other than that I don't count that any action has actually happened until it has fully resolved, because 5e is chock full of "take back" mechanics where an attack hits and you use a reaction to make it so that you blocked it and it didn't really hit, etc.
How do you quietly stab someone? How do you declare an action to kill someone in their sleep? These are things you can try to do, but then those actions are resolved using the combat mechanics. (At least, they are in my game.) Other editions of D&D (maybe just one edition) have rules for insta-killing a creature you've gotten the drop on. It's an assassin class feature. The closest thing 5E has is Assassinate which is a bonus to attack and damage rolls conditioned on winning initiative. If you roll well enough, and your target doesn't have too many hit points, your attempt to quietly insta-kill them may indeed prove successful, but just declaring that's what you do doesn't really fly in my game because it's the DM's job to determine if your attempt is successful or not.
I think how you're describing initiative is a big difference. The attack roll and "take back" mechanics are part of the resolution of an action that has already been declared and committed to by the player.The fact of the attack happening in the fiction isn't generally dependent on the resolution of the attack unless we're talking about an ability to actually alter events that have already happened which the shield spell, for example, is not.