Disagree on both these points. Yes if you walk up to stick man and make a scene threatening him with bodliy harm, that it is true. But that is the same if you walk up to stick man and try to bribe him (persuasion) and people see you or it fails and he tells. Yelling at someone and threatening to stomp them into the ground is not the only, nor I would argue generally the best way to "intimidate" someone.
Sure, if he pays him off, but what if he simply appeals to the guard's better nature to help some poor travelers who don't have the coin to spare? That is also persuasion and you aren't breaking any laws to do it.
As for the second, absolutely you can intimidate all of those powerful beings and I have a good example from one of the Richard Lee Byer's FR books: A good cleric of some sort get's a dracolich's phalyactery (both and ancient dragon and a lich, two of your examples) and successfully intimidates him into helping them for quite a while ...until the deracolich comes up with a plan and gets it back.
Now if you are saying you can't walk up to those characters and show them your muscles and intimidate them you are right, but that is not the only way to intimidate.
... Would you even have them roll for that? I mean "I am literally holding your soul in my hand and have no problem ending your immortal existence" is less abotu intimidation and more about the fact that you got a Dracoliches Phylactery. I think that is a situation where you can no longer fail, so no roll even happens.
Which kind of misses the point of using the skill.
1) Threaten the vampire lord with consecrating his last remaining coffin by casting hollow on it ...or draw a magic circle, tell him it is a teleportation circle linked to the middle of the ocean or running water and threaten to shove him into it (combination of deception and intimidation). 2) Tell the church of light cleric you will sacrifice 100 captive townspeople to Grumsh and eat their children if he doesn't do what you want. 3) Threaten to assasinate the old emporer's son, his only hier and his legacy. 4) Threaten to bring the order of the guantlet down on the warlord. All of these are examples of intimidation and all could work against those powerful NPCs you mention in the right circumstances.
1) Again, you've already won by the point you have access to the vampire's last coffin, or can seal him in a magical circle. Plus, as you said the cirlce is far more deception than Intimidation anyways. So... yeah, go ahead and use your skill proficiency to convince someone you already defeated and have under your power to do something... that sort defeats the intent of me saying you can't use this skill on someone stronger than you. If you have to have them at your mercy before the skill even applies, then it is a lot weaker than Persuasion or Deception.
2) Congratulations, you are all heretics worshiping dark gods. You may succeed, but you will have paladins and crusaders and maybe even angels hunting you down to remove your evil from the world. Or... you could have just tried to use persuasion to convince him your cause was for the greater good and not gotten an entire inquisition sent after you.
3) Congratulations, you now have an emperor who is seeking your death. If you don't already have his heir at knife point, the entire royal guard of powerful knights and wizards will begin trying to kill you. If you do have him at knife point, you will be visited by whichever assassins an entire Empire's treasury can hire. If he is a really vindictive old man, he might even hire extraplanar assassins to kill you. Also, this assumes the Emperor only has a single heir, and that he is weak enough you can capture and hold him prisoner.
4) No idea who the Order of the Gauntlet is. This guy is probably already on their radar though, so at best this is an empty threat because he is already dealing with them, and at worst... well he just kills you all to prevent you from leaving to go tattle on him. So... no, I don't think that is going to work.
One helpful thing to notice is that the skill intimidate has the main use of "coercing" people -- making them do things they don't want to do. That's a clear distinction from persuasion, which changes what people want to do. Coercion doesn't need to be an obvious threat -- it can simply be the knowledge of power. When a level 15 fighter tells you to get out of the way, they don't need to threaten the regular guy; you know they can kill you without breaking a sweat. I'm sure you can think of many movie scenes where people are terribly polite while they coerce people to do what they want.
It's also used in between-sessions time to manage employees; if you have a bunch of hirelings building a castle for you, intimidate is an excellent skill for judging how well they work.
Two points
1) Does a level 15 fighter even need to roll? Again, I don't really find it compelling to tell me the skill is meant for my level 15 dragonslayer to tell a potato farmer to step aside. I'd like to think I can do that with no problem.
And this gets right back to the issue, if Intimidate is only useful against people far far weaker than me, or whom I have a sokid undeniable advantage over (like literally knife to their throat) then what use is it? Am I really going to have the rogue roll to intimidate the put purse while he is millimeters from stabbing the guy to death? The only thing the guy can do is call the bluff, and then the player either kills him (losing the information) or backs down (losing the information) and if I succeed... well this guy isn't exactly going to sing my praises for threatening to kill him
2) Intimidate your employees? Really? Imagine you have a boss who constantly "coerces" you into working. Do you think you are going to be happy working their? Think you are going to be loyal? I mean, this is straight out of the Dark Lord's Handbook. You don't go around threatening the help, that just incentivizes them to hate your guts.
I wonder if that is an issue because GMs are not roleplayingt the consequences of diplomacy?
Sure, you can sweet-talk the guard into letting you pass, but you've now left behind someone who feels maybe betrayed or faithless. Or maybe they get greedy and come at you later, asking for more bribe or they'll turn you in.
As a GM, I feel I should be providing consequences for everything the players have their characters do. Players should be choosing between "two tactics with different consequences"; not between "one tactic with no consequences and one tactic with bad consequences."
That way intimidation and persuasion simply become "two tools for overcoming an encounter", not "Intimidation is the wors skill ever."
Why does everyone associated Persuasion with Bribery? Most of my PCs never bribe anyone, but we sure do a lot of persuading. And you aren't tricking them, like you are with deception. You are literally convincing them to agree with you, so how is that a betrayal? "that nice man in the coat made many good points and I agreed with all of them. How dare he treat me this way!" that doesn't make any sense.
I wonder if a lot of people run Persuasion way differently in their games, if they think of it as bribes and lies.