From this point of view, the reunion of such branches is a spiritual imperative; hence a certain sort of ecumenism, for branch theorists, is a moral imperative. But there is a very good reason why their kind of ecumenism has a great deal of difficulty gaining traction: it does not square with the self-understanding of either the Catholic or the Orthodox churches, and is a minority view even within the Anglican Communion. Branch theorists know that, of course, and respond by arguing that their ecclesiology is more firmly grounded in the actual doctrine of all three communions than the prevailing self-understanding within those communions. The apparent gnosticism of such a view is by itself reason enough to reject it; but merely pointing that out is not an argument. My purpose here is to adumbrate such an argument briefly, in two steps.
On the positive side, consider the fact that, in its dogmatic constitution on the Church, Lumen Gentium, the Second Vatican Council famously asserted that the Church of Christ "subsists in" the Catholic Church, i.e., the communion of churches in communion with that of Rome. That was a verbal departure from many previous magisterial affirmations that the Catholic Church just "is" the Church of Christ. But employing that same "hermeneutic of continuity" which has been employed since Vatican II by the present and the previous pope—who were both present and active at that council—it is not difficult to explain why the verbal departure is only a development, not a negation, of the older doctrine.
The Catholic Church has always regarded the "Eastern" and "Oriental" Orthodox churches as "true, particular churches" with apostolic succession and valid sacraments. She has also insisted that she is "the" Church. How are those affirmations to be reconciled? By pointing out how it's possible that the whole can exist without the full integration of some of the parts. True, particular churches "belong properly" to the Catholic Church as parts of her which, in the case of some Eastern churches, are unduly detached from the whole. Even if such parts are in "imperfect" communion with her due to schisms brought on by sin on all sides, the whole OHCAC "subsists in" the Catholic Church inasmuch as she continues to exist as a subsistent whole therein, though wounded by the partial detachment of some of her parts, which in turn retain significant "elements of truth and sanctification" belonging properly to OHCAC and thus to the Catholic Church.
From a Catholic standpoint, then, there is no basis for branch theory. For the alternative picture presented by branch theory has it that OHCAC does not at present exist as an integral whole anywhere, which is incompatible with the doctrine of the Catholic Church. Branch theorists hold instead that the detachment of true, particular churches that are proper parts of the whole means that the whole itself no longer subsists, and thus does not continue as a subsistent whole, but is rather a collection of parts called "branches." The idea that OHCAC exists as a collection of visible communions rather than as an integral whole in one, visible communion is what allows branch theorists to insist that their view does not undermine the traditional dogma of the Church's "indefectibility." But that brings me to my other, negative reason for rejecting branch theory.
Branch theorists such as Fr. Matthew Kirby of The Continuum like to argue that, by the theological criteriology of Catholicism itself, the Catholic Church has never "infallibly" taught that OHCAC subsists in the Roman communion as an integral whole. They also like to argue that the Roman communion cannot be OHCAC because, even though OHCAC as a whole is infallible when teaching on matters belonging directly to the deposit of faith, the history of Catholic doctrinal development shows that Rome has negated certain doctrines that, by any fair criteria, would have met the criteria for having been taught "definitively" and thus "infallibly taught by the ordinary magisterium" ('ITOM') for short. Last week, Kirby and I debated the question whether Catholic development of doctrine on the punishment of heretics is an example of such a negation. I of course maintained it wasn't, and Kirby maintained it was. You can judge the result for yourself by following the combox.
On the question at hand, the procedure is reversed. Kirby has made clear that, according to him, the doctrine that the Catholic Church "is" OHCAC has never been either solemnly defined or ITOM. I of course would maintain the opposite. I don't have time to do that here. But I want to end by reminding readers that branch theorists have a very long row to hoe. They have to show that the way they apply Catholic theological criteriology is more reliable, by Catholicism's own standards, then the way the Catholic Magisterium itself has been applying it. That's what makes their view what Newman called "private judgment." And that's why it can't possibly be Catholic.