Tha analogy still fails due to the fact that no other country in the world, and certainly not the USA, recognized the legitimacy of the CSA. However, nobody in their right mind would refuse to recognize the legitimacy of the Orthodox or Southern Baptist denominations as existing Christian denominations.
As for whether Orthodoxy split off from Roman Catholicism or vice versa, clearly the two sides disagree on that issue.
Your refusal to admit that the Pope is not the head of these other denominations is as nutty as if GWB refused to admit that Tony Blair is the Prime Minister of the UK.
None of the ancient sees established by the apostles survive unbroken to this day except for Rome, the ancient see of Peter. All the patriarchates are beneath Rome in terms of authority as any perusal of the first three hundred years of Church history easily shows.
The Orthodox Churches actually admit that, of the 5 patriarchs, Rome's is first among equals. However, the Orthodox Church simply does not admit that the Pope is the head of Christianity, and neither does any other Christian denomination.
Your claims fly in the face of reality.
You don't seem to understand. The presence or absence of another country recognizing the CSA presupposes that the country doing that is both really a country and really competent to make the judgment.
Now, there is only one Church. Jesus did not establish multiple churches, just one, all part of one body, His. So the analogy is not perfect because we live in a world with multiple countries, but only one True Church.
Any group which has valid apostolic succession (like the Orthodox) thereby has seven valid sacraments (like the Orthodox) and draws her power from the ONE Church headed by the Pope. If that group refuses to communicate with her head, she has rebelled, broken off, as Lee did against Lincoln, yet she remains fully Catholic in all but obedience.
Other groups which broke off without benefit of keeping valid apostolic succession (like the Lutherans and all their off-shoots, or the Anglicans) are simply ecclesial communities, a bunch of lay people who have only baptism and marriage as sacraments and are too at a loss for grace to fully recognize even those remnants of grace that come to them through the power of the Catholic Church.
Your claims fly in the face of reality
Your opinions are, fortunately, not reality. :)