Seriously. It's a good example. But how would you adjudicate this scenario:
Bob: I swoop in. I've got my ring of invisibility on and my ring of silence.
DM: Alright. The guard doesn't see you. He goes back to reading his magazine.
Bob: I watch him for an hour
DM: He just sits around. He dozes for a bit. Reads his magazine some more.
Bob: So there's no indication he knows I'm here.
DM: No. Your stealth check passed his Passive Perception.
Bob: Great. What's he doing now?
DM: He's gone back to reading his magazine.
Bob: Excellent. I want to throw my invisible and silenced dagger at him.
[AT THIS STAGE THE GAME SWITCHES FROM NARRATIVE MODE TO COMBAT MODE]
DM Roll initiative. He's surprised.
Bob: Damn... a 9.
DM: 10! He wins!
What happens next? The act of throwing this dagger will not cause it to become visible or audible.
An extreme outlier, a DM would be within his rights int this situation to rule the inept guard cant possibly react to an invisible assasin throwing an invisible dagger. Thats what DMs are for after all.
That's peachy and all. But those aren't the rules I asked for. I saw those rules (I even mentioned them in the paragraph you quoted). Can you please direct me to where the rules in the PHB say "The ONLY way combat can be initiatied is when someone declares a hostile action". As I mentioned I can't find them.
Play changes from loose narrative play to the structured combat sequence when combat is initiated. That occurs when someone (either the DM or a player) declares a hostile action. Attacks happen in initiative order during combat.
When you see an Ogre in a room, you dont need to roll inititive to speak to it. You only roll initiative when hostilities break out. There are no free actions for people that surprise other people with hostilities. Actually there technically are free actions when people surprise other people or attack them from hiding or ambush them - the rules for such surprise attacks are in the PHB combat section under 'surprise'.
Bob says "I want to shoot my bow" at the NPC. Bob then wins initiative, does he get to change his mind and decide not to shoot the bow? Bob says "I want to shoot my bow" at the NPC. Bob then loses initiative and acts dead last. Does Bob get to change his mind and decide not to shoot the bow?
Bobs player has declared a hostile action. This switches play from narrative mode to the more tightly structured round by round/ turn based abstraction of the combat sequence. This was a decision that Bob made. Bob (the player) has total agency here remember.
If Bob wins initiative I assume he fires his bow. Why wouldnt he? In between him declaring his action, and his first turn in the combat, zero time has passed. I would certainly not let him 'take back' his declared action (I might though). If I did, then no harm, no foul. If he's hidden then initiative is ignored and narrative time continues. If he was in plain sight of the enemy, he's probably nocked an arrow and taken aim at a NPC so they would react accordingly (and initiative would probably be retained, and he may find himself getting attacked depending on the NPC).
If Bob only said he 'wants' to shoot his bow, nothing happens. There is no initiative roll. Wanting to do something isnt a hostile
action. If Bob declares his 'is' shooting his bow at a NPC (i.e. he nocks an arrow and aims at someone, drawing back the string) then initiative is rolled, and the enemy get a chance to act in time to his trigger (via initiative) either ducking into cover at the last second, or whatever.
This is the logical disconnect.
The assassin has already beaten the guard Stealth/Perception-wise. The guard doesn't know the assassin is there.
How can a Dexterity check let you hear something that your Wisdom(perception) check didn't?
Youre narrating the abstraction of combat. Maybe the guard just gets a 'bad feeling' or luckliy ducks down to reach for his packet of smokes at the last second. Its up to the DM to narrate how it goes down.
Dont get me wrong, I'd narrate it differently if Bob had the skulker feat (the assassin would be unaware of Bob if Bob misses, so he narration would occur after the attack is resolved). In that case I'd narrate it as the inept guard moving at the last second (ruining the assasinate attempt). Or course, even with the skulker feat, if Bob hit the guard, (and he obviously needs to in order to trigger assasinate anyway) then Skulker aint gonna help one iota, and the Guard is fully aware of his presence (the crossow blot sticking out of his shoulder is evidence enough).
Its all an abstraction that is to be narratted accordingly. Just like hit point attrition (another abstraction). When you 'Hit' someone with a sword (via a sucessful attack roll) you dont always actually connect with them. They could have dodged at the last second (taking 'hit point damage') parried it (taking 'hit point damage') it glanced off their armor (taking 'hit point damage') or whatever.
Combat is an abstraction. From turn based activity (youre not actually standing around waiting for one person to move 30' and then attack, then another, then another etc - its all happening more or less at the same time) to hit point loss (most 'hits' are not actual wounds or even actually someone getting physcially struck) etc. Just narrate it accordingly.