Earlier quoted context omitted.
What happens when Twitter crashes and burns, and now you can't access your blog?
I would bet a small amount of money that Twitter outlasts Medium. And if Medium outlasts Twitter, do you think Medium will just sit by and say "well that sucks that none of our users can access their blogs"?
In that case, think of a 24 hour datacenter outage then.
Unless Twitter gives Medium more information about usera on authentication than I think they do, I don't know how Medium could possible verify the owners of each blog.