The Enclave isn't supporting Vault-Tec at the time of the meeting because, bluntly, what we're seeing there is almost certainly the BIRTH of the Enclave.
It is the birth of the Enclave in some sense that it's making itself known to business and manipulating them into joining it, but the show pretty clearly telegraphs with the shadowy g-man watching the meeting that the Enclave already pre-exists as a distinct form unto itself, pulling the strings from the shadow. Because the show is written by and/or for simpletons.
Children of the Cathedral
The Master
You're cooking the books dawg, you just counted the Unity twice
I suppose you probably meant to say "The Enclave" here for Fo2, since it's followed by "The Enclave again"
How does Mr. House constitute a conspiracy? There is the fact that, by the strict letter of the law, Mr. House can't conspire because he drafts his plans entirely alone.
Conspiracy, con meaning together, as in with others. That's a nit-pick of course, but putting it aside, I don't think there's anything particuarly "conspiratorial" in the every-day sense of that word about what he's doing. He has a plan, but that's not really the same thing.
I'd say a problem with your comparison to these various antagonists with the exception of the Institute (and possibly the Enclave as it's depicted in Fallout 3 - there are fan speculations that the Enclave is responsible for destabilizing the Capital Wasteland and doing things like paying Talon Company to take out do-gooders, but that's just speculation to justify poor worldbuilding), is that while these are groups that are conspiracies or conspiring to some degree, they're not really responsible for the setting as it currently exists. They're groups that have aims they try to achieve in secrecy, but they're in the process of trying to make that happen. Sometimes they win, sometimes they lose, but they're not these uber-powerful puppet masters who mostly get what they want and shape everything you see.
The closest you get is with House, given he is responsible for the setting of Vegas not having been nuked largely, but he bumbles his way through this and is constantly getting owned - He didn't get his chip on time, a few nukes got through, his computer system got fucked, he fell asleep for a hundred years and when he woke up was basically impotent for another hundred until an opportune moment, bumbled his way into picking tribes for somewhat arbitrary reasons and ending up with a tribe of cannibals who are actually willing to go back to cannibalism behind his back, a tribe of traitors who (Imagine My Surprise) seek to betray him, and his hand-picked protege tried to fuck him over and take over Vegas for himself. He's a competent guy, but he bumbles his way through things as anyone would.
The way it feels like Vault-Tec is presented in the show is more like an omnipotent controller of how events transpired, sort of like how (I'm told, haven't seen it) Palpatine is presented in the Rise of Skywalker, "I am the puppet master, I've been pulling the strings, I'm responsible for everything!!" as sort of a lazy ass-pull.
In the interest of fairness - My argument will be undermined somewhat when, as I currently expect, next season or so it's revealed that while Vault-Tec
intended to drop the bombs but were beaten to the punch. Still, I think the argument above holds water for the spirit in which they're depicted, if not the letter of the plot.