I think the basic idea is that each slack organisation is it’s own independent thing. Sure, they all use slack, but each instance of slack is it’s own individual instance, and as such has no contact with the outside world.
Think of is the same as Discourse (the software which powers this forum). You’d not expect to be able to use your login details from birmingham.io to log in over at https://meta.discourse.org/, and it’s the same with Slack.
I know this is different in that slack hosts all of the instances, but so can discourse, and again, you’d not expect to be able to use the same login for all the instances of discourse hosted by discourse.
I don’t think the issue here is design, but rather communication of intent, or education in the use of the system. I agree that it’s confusing, seeing as they’re all called Slack, but once you understand the principle above, it should make more sense.