They knowledge about Mab guarding the outer gates is not widespread in the white council otherwise their stance towards the winter court and the Sidhe in general would be different.
I'd say all of the Senior Council, and some of the senior Wizards like Klaus the Toymaker i.e. just about anyone considered for Senior Council status would know about the Outer Gates, and at least part of their purpose. Many of them have been life-long scholars and their lives are a few hundred years; they probably picked up a few things.
Would the White Council treat the Winter Court differently if they all knew their real purpose? Perhaps, but I think it would largely be unchanged. Particularly in matters of foreign policy. At the end of the day, despite Winter's purpose, they are still largely a horrible murderous bunch that like to prey on humanity. I think the White Council would still treat them with heavy suspicion and limit their influence in the mortal world where they can, and defend mortals as much as they can from them (although the latter part is pretty much Summer's purpose). Look at how Ebenezar treats them - he is respectful but keeps them at arms length, same for Listens-to-Wind. Ancient Mai seems to be on reasonable terms with the Fae but I doubt she trusts them. The Merlin seems to have a fairly clinical view of everything in general. Cristos seemed to like to build political connections, so I'd say he would be friendly (at least openly), but like most politicians I doubt he has any real affection for them. Martha Liberty would be the hardest to judge but she seemed quite formal and respectful with Mab, so I'd say she's about the same as Mai or the Merlin. The Gatekeeper obviously has a fairly close relationship, you could even go so far to say that he has a 'good' relationship with all the Fae, but that's probably just as much to do with his position as his personality (he's a bit of a charmer).
Myself, I think they are about the overturn of the old order without full knowledge of what that order is for- with a few nihilists and anarchists tossed in.
Take the Outer Gates. Wizards like Eb know what Outsiders are, but do they know that the Winter Court is responsible for keeping them outside the Outer Gates, or do they just think the Gates are "shut" instead of being a citadel under constant siege. If they have that misapprehension about the Gates- that there is no need for them to be defended- then there would be much question of the Accords.
I've brought this up before, but why the Accorded nations? It makes the most sense to me if they are a mix of B-string and heirs to Winter's duties on the Gates. With the human population exploding, Red or Black Courts could take over manning duties, for instance. I've mentioned before too that I am partial to the idea that vampirism "plugs into" the port Outsiders would need, making vampires immune to direct control by Outsiders (Though not willing participation).
So what is the Black Council for? Is it a wizard-centric organization that takes the passivity of the White Council as unacceptable? Is it a union of beings looking to overturn the old order to forge a new Accords in ignorance of the threat to reality? Is it a collection of Nemesis pawns- or some combination of all of these?
Hints in the series indicate there's some kind of astral cycle that at some point, only starborn can do something, and it's a time of change- so maybe it's those planning ahead on the cycle. Maybe it's no more than a temporary union of disparate players aiming to control starborn.
The distinction between Gray Council and Black Council is based on Harry's trust of Eb- if I recall correctly, a careful parse of Changes reads that the Council showed, and also Vadderung showed, but that doesn't mean they showed together. If they messed with Kukulkan, or otherwise were traceable to the success of the Red Court, they eliminated that evidence by backing Harry.
Definitely seems to be the theme of the Black Council - disrupt the status quo. Could be that they are merely anarchists or even nihilists. But from the agents we have seen I would say neither in truth. They seem to have a set purpose, and most of it is about gathering power whilst depowering and disrupting other groups.
My theory is that Jim hadn't really planned how old the Accords was, and then sort of changed his mind half-way in. Narratively speaking I'd say they mostly serve the purpose of gathering and showcasing the various supernatural groups. But in-universe, as they were a fairly recent creation of Mab's, I'd say they have to do with the coming apocalypse and the B-team idea is as good a reason as any. Although it might be simpler than that: Mab is gathering every ally she can just to hold back the tide. She hasn't really got a plan for 'after' because there might not be an 'after'.
I think you're onto something about the composition of the Black Council and their goals.
That's a very good point - the real difference between the Black Council and the Grey (from what we currently can see) is that one is lead by supposed good guys. But Eb doesn't have a stellar record either, and neither does Harry or Vadderung. I am not sure what section of Changes you are referring to - but when Lea summons the Gray Council, one of the twelve figures is Vadderung. So I'd say he is on that team. But Harry does point out to Murphy that while it wasn't officially the White Council who showed up, it was what the White Council was supposed to be. Considering Vadderung taught the original Merlin, and that Merlin created the White Council, I'd say there's a good chance he is trying to start again. I doubt that Merlin was a white knight either, he hardly is in the original tales, and Harry points out rather wryly that if Merlin's all over the pages of history he probably was some kind of bastard. The same could be said of Harry for that matter.