> As a public institution you, the citizen, own it.

Nominally, yes. In terms of that meaning anything, no. The benefit of ownership is not exclusivity, but control. If the library doesn't have a book (or other piece of media, of course), I have no power to influence them to get it despite that theoretical ownership. If the librarian decides a book is offensive and removes it from the collection, I have no power to influence them to keep it. I have to live with someone else's decisions about what the library does and does not contain, just like with a commercial service. So my nominal ownership really means nothing at all.