Your Protestantism is showing. The hierarchal nature of the Church is not a governing policy anymore than the skeletal system of the human body is a person's choice as to how to run his body. I'd be interested is seeing the Swedish equivalent of the Act of Supremacy, the only other Protestant body to recognize this fact as the essence of the Church.
I do not see the accuracy of your analogy or even its relevance, and if you are going to point at hierarchy as the necessary mark, then the Roman have it and you do not. Or to put it in other words, you are making a claim to unity which is not realized in the world, especially considering the lack of communion among all the churches represented by their members in this forum who call themselves Orthodox.
It would be far better to abandon all the condescending analogies and deal with the worldly reality. Once you are claiming Christ as your head and refusing discussion of how that headship is realized in the world, you are making nothing more than a metaphysical claim which all churches make.
Trying to bring the Church down to this?
I might as well accept the Muslim claims of being the true Christians.
I brought up the Swedish Church to make the argument less Anglocentric, and the worldly reality is that the Swedish Church is the best example of another Protestant body that claims Apostolic succession more clearly. There are others (now, for the most part, conviently lumped together in the Porvoo Agreement), but most of them being Lutheran, and the Augsburg Confession/Book of Concord itself does not have a well defined dogma on the episcopacy, the dogmatic statements on this matter leave much to be desired: the historic practice of the Swedish Church, like the Anglican, fills in some gaps. One might through the Methodists in (who claim apostolic lineage, oddly enough, through a visiting Greek bishop), but since Wesley never claimed episcopal consecration (he held presbyters could do it), that is not without its problems. These Protestants (and various vaganti) basically sum up the Protestants who claim Apostolic succession. Have I forgotten anyone?
Of course, then there is the One, Holy, Catholic and Apostolic Orthodox Church (both EO and OO), the Vatican (which in reality has a bishop, not a hierarchy, but is close enough) and her daughters (Altkatholisch, Polish National), and the Nestorian Church(es). Have I left out anyone here?
So back to "metaphysical claims all churches make": No, the pictures of each ecclesiology is NOT the same, and it is a mistake to say it is. Or rather, it is a stacking the deck in favor of Protestant eccleisology, and Radical Reformer Liberal Protestant at that. Those "churches" which have no apostolic hierarchy exercise no headship in the world. None. There claims are as valid as the Muslims saying that they believe Jesus is the Christ, the Bahai's claiming Him in the succession of their continuing line of prophets, or the Mormons claim to restore His Church. No hierarchy tracing back to the Apostles, no Church. And no Church, no Christ. Period.
Among the Protestants who claim Apostolic Succession, the Anglicans place the greatest store in it, perhaps matched by the Swedes and Finns (their primates would not take part in the installation of the ELCA presiding bishop, because of the issue of apostolic succession). The Finns, with their Church Act, may be the closest to this
But they all have the problem that their statements lack precision (It would be nice if thishttp://www.porvoochurches.org/whatis/resources-0201-english-4.php
could be a basis on which to have a discussion). So the Anglicans et alia don't like the image given herein. So they say they depend on the Book of Common Prayer, the problem that it is not Common across their "communion," would result in a body with a book for a head, and is a rather odd position for a church that holds "sola scriptura" as a dogma. The problem with all the Anglicans and Lutherans is their "apostolic succession" came by way of Erastianism. That is the historic, real word reality: headship was realized in the world through the crown (this is true of the other Lutheran churches as well).
(btw, as a side note, the Orthodox primate of North America, Bp. John, when the US took over secular control of his diocese
Bishop Johannes, of the Russo-Greek Church on the Pacific coast, has ordered the prayer for the President of the United States, contained in the Liturgy of the Episcopal Church, to be used by the Greek Priests. http://orthodoxhistory.org/2010/07/prayers-for-the-president/
this was to fulfil the apostolic command to "pray for the emperor," not as part of the apostolic succesion of the Church. But the PECUSA's BCP came about because the church had to be cut loose of the British king, who had already given the US its independence. And then the Americans had to go to the nonjurors, a group of the church with no basis of distinction in the church but in politics).
So that leaves the so called "Apostolic Churches," so called because of their valid historic claims (whether they are theologically valid as proved by history is a different question, here put aside). And the pictures for them do not match. The OO and EO are identical, as, on this issue, can be said of the ACE. Perhaps the Altkatholisch can be said to be here too, though the Episcopi vagantes cast that into doubt. The PNCC could be seen in that picture. The Vatican's view, however, cannot.