Elegast,
I'm with you on nearly every piece of this, but I had a different take on the Blackstaff.
My mind took:
MW's missing walking stick,
WOJ on the Blackstaff preventing insanity,
Outsiders/Nemesis' tendency to cause insanity,
Winter's current purpose (defending against Outsiders), and
Knnn's theory that the Laws of Magic exist to prevent Outsiders from gaining influence...
...and cooked up the FrankenWAGTM that the Blackstaff is specifically a defense against Outsiders' ability to exploit openings made by violations of the Laws of Magic (among other things).
The Blackstaff would seem alive because it's pulling the Outsider influence out of the user every time they're open to Outsider infestation. Note that the Blackstaff's tendrils pulling from Eb's arm are reminiscent of Mordite, and the anti-life feeling from Mordite is in turn very similar to the lack of soul in Harry's hand after his first use of Soulfire.
If this is the case, I doubt anyone on the WC knows MW's walking stick's true purpose (i.e. allowing Death itself to walk freely among the Outsiders without being vulnerable to their corruption), save maybe for Rashid. If Rashid had ever confronted Eb about returning the Blackstaff, that could explain the apparent breakdown in trust between them that Eb refers to in PG*.
Bonus factoid for MW's walking stick = Blackstaff in general:
This would mean MW would have extra cause to be upset by a mortal harming her by summoning her when she can't travel- the mortal who summoned her is the grandson of the one holding her walking stick!
*Also, in a rare double-whammy post, I don't believe that Eb created the Grey Council sometime between PG and TC in response to his conversation with Harry at the end of PG, as others have posted elsewhere. At the very beginning of PG, Eb tells Harry he's already begun organizing a group of people he trusts- I think the GC predates Harry telling Eb about the BC theory.
Lastly, if Mother Winter is Death, does that make Mother Summer Taxes?