Nor can we just use a Great Deed from the Emperor to get the Jade College to share secrets, just as we couldn't use a Great Deed to obtain the secret to Gehenna's Golden Hounds from the Gold College.
Yes, this exactly. Even if you
technically have the authority to, you cannot
effectively order somebody to give up a secret unless you already know that secret yourself. Because otherwise it plays out like:
Emperor: Hey Jades, I officially order you as Emperor to tell the protagonist about the cool magic thing that you apparently keep secret, I guess.
Jades: ...Of course, Emperor. Though that is, arguably, a grave overreach! Some might say. Not us, of course. And certainly not any other of the Colleges, or Cults, or anyone else in the Empire who might have organizational secrets they wouldn't care to be ordered to hand over to pay off an unrelated political debt of the Emperor's. Anyway, here you go protagonist: [tells Mathilde nothing or almost nothing she doesn't already know]. There you go, we did the thing. *maintain poker faces*
Emperor: ...Okay, then. Good. I don't know what the secret was, so I have literally no way of confirming that you did in fact share everything you know. Which means I can assume you did with a clear conscience, since I have no feasible alternative anyway. Great Deed paid off! Damn, it feels good to be the Emperor.
Jades:
We're gonna fucking remember this, protagonist.
Emperor: What was that?
Jades: We can honestly say that it's nothing you need to worry about.
Emperor: Kay cool then, I won't. I can't wait to get back to looking forward to the absolutely no long-term political consequences on the institutional attitudes of the Colleges towards the Emperor that are definitely what's coming.
Anyway, here's my argument for Theologian: this is the Deep Lore option. This, right here. Delving into the secret mechanisms of divinity itself?
Hnnngh. Do want. Plus, just consider: we could develop an internally consistent, logically grounded, basically irrefutable really, thesis that objectively proves that Sigmar is actually a big ol' turd that people shouldn't bother worshipping.