Persona solves the problem of uniquely naming you to the site, on the assumption that you already have a unique email. There's more pieces here with their own issues, but I think Zooko's triangle is still relevant.
"Actually, I don't think there's anything wrong with nicknames and their lack of global uniqueness. Moreover, I believe it's a good thing."
That depends entirely on the applications you're intending to put them to...
That's irrelevant to whether Zooko's triangle applies, which was all I was weighing in on.
That said, yes, consumers totally have this problem. We encounter it visibly every time a username we want is taken when we try to sign up for a site. The effectiveness with which Personal eliminates it entirely, ameliorates it, or merely pushes it off to other parts of the system depends on other details of the service in question.
It's also not the only problem Persona purports to solve, simply the relevant one.