This idea of the model and its extinction perhaps refers to the Balamand declaration,
east2west.org/balamand.htm,
which states:
" With regard to the method which has been called “uniatism,” it was stated at Freising (June 1990) that “we reject it as a method for the search for unity because it is opposed to the common tradition of our Churches.”
Balamand does not give an explicit definition of “uniatism” Just what it is that is being rejected is unfortunately left open to interpretation and thus to polemical exaggeration. However, the basic point is this:
"Because of the way in which Catholics and Orthodox once again consider each other in their relationship to the mystery of the Church and discover each other once again as sister Churches, this form of “missionary apostolate” described above, and which has been called “uniatism,” can no longer be accepted either as a method to be followed nor as a model of the unity our Churches are seeking. "
Setting aside discussion of the merit of historical charges and counter-charges, we simply are seeking cooperate reunion; we do not proselytize each other’s people or particular churches.
There is a nice article from a priest of the BCC that addresses the historical problem embedded in the statement, while providing an interesting outlook.
findarticles.com/p/articles/mi_hb3236/is_2_34/ai_n28699493/?tag=content;col1
I am curious about the references of course. In reality, far more than any model of unia, it is the Balamand declaration that seems extinct.