Access is not control. They can approach and talk to people, up to the President, but they cant give people orders, nor do they have the kind of statutory or informal authority necessary to prevent said people looking. Nor can they police other agencies for infiltration. Thats kind of the problem here.
I'd absolutely approach them if I didnt care about who knew it. But we do care.
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Von Triers is a senior agent, not their leader as far as we know. We dont have contact with their actual leader or governing board. And the Federal Reserve incident was evidence that no, she doesnt have those kinds of personal contacts, presumably because the Library's remit has been to keep a low profile, something that was reinforced during the Cold War.
Compared to, say, Rashid who personally had the contacts to clear any legal issues from Harry returning from the dead in Cold Days.
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Its not going to be secret to everybody.
But since the bad guy factions are not a monolith, being able to restrict or impede the spread of information as a priority will materially affect the ability of bad guy factions to make use of this knowledge.
Which is why I'd rather not work through the Library here.
They don't need control for something like this. We're not talking about anything illegal, or arguably even unethical. This is essentially an asylum request from people under active threat of persecution.
A trusted authority reaching out to a personal contact and going "yes, we can confirm from an ongoing investigation that these guys are related to someone the Cartels want dead. Fast track and keep it quiet till they're in country" would speed things up and stay low profile for long enough that it doesn't matter if done correctly.
The Library's job is to know about infiltration and work to stop it, and failing that to work around it to protect the country. They should be able to do stuff like this.
Von Trier was the person in the room with the Senator running the Vegas incident, she was the one who would have been briefing the president if it went sour. I have no idea what you mean with the Federal Reserve thing, because Von Trier got the exact right person on the phone with us at 2am to handle the problem. We had to explain the problem, but it's not like Von Trier had time to pick up everything she needed to do it herself before the deadline anyway.
Being able to make that call happen is a sign of serious connections, and choose to leave it to use of good judgment.
I don't understand why you think the incidental contacts of older wizards outweigh the contacts of people whose lives revolve around the system. Rashid doesn't spend all day dealing with the US government, he has shit to do. Being magically powerful doesn't suddenly mean you're omnicompetent.
Beyond that, for Von Trier to be as old as she is she must be either a freak of nature grade sorcerer or a wizard level player. Being over three hundred years old and still looking late middle aged doesn't just happen without that or some kind of deal in the DF.
From what we've seen of her she's built at least part of her skill set for political/social applications. She picked her strategy for convincing the Senator by looking at the future and picking the path he'd find most persuasive, for example. That's not a minor trick in this context.
Even discounting the Library at large, which we shouldn't, a wizard who spends her entire career working with an organization and bends her magic to aiding that purpose is going to have better access than an external party who doesn't.
The council even had their ambassadors to the US work through the Library, better than even odds McCoy or Rashid's first call for this stuff will essentially be to her.
The key here is to get this done before people can react, any bad guy who can learn about this from our movements to bring her in will be able to learn it from her arrival. The critical security period is the transition.