Earliest church model; who has it?
Just as Mommaduckofmany’s post clarifies that the British Isles were evangelized well before any major schism, so it is not correct to say that they were evangelized by “Catholic” and not “Orthodox”, I would say something of similar of ecclesiology: It didn’t “belong” to anybody. It was simply the way that the early, undivided church functioned.
When I look at the Orthodox church, that is where some of the confusion for me comes in. Let’s take the New World. Would not the earliest church model dictate that missionaries with the gospel come over, preach, those that believe form their own congregation, leaders are then taught and put in place from that local area. Is that not the claim of the Orthodox church? So, why is there a “Greek Orthodox Church” in say, Boulder, Colorado. Why is it not “The Church at Boulder” with Boulder natives in the church and leading it?
It is, but there is more to this question than you realize. In addition to the already posted answer that many were refugees rather than missionaries, there is also the fact that for a specific rite or usage to develop naturally in a new society can take a long time, particularly in the modern world where there are so many different options and influences out there. In the old days, when there were only Orthodox Catholics and various heretics (at least so far as Christianity was concerned), some traditions grew to be predominant in a given area (e.g., Coptic and Coptic-influenced churches from Egypt and Libya to the Sudanese kingdoms and Axum), but even in the cases of those traditions, they are themselves developments. We know that St. Mark originally brought Christianity to the Greek-speaking city of Alexandria, and therefore the liturgy of the Church of Alexandria was originally in Greek. The earliest surviving bilingual Greek/Coptic Bible fragments date to about the middle of the second century, or just under 100 years after the martyrdom of St. Mark in Egypt, in 68 AD. So this is a sign how fast the faith spread in the country, but it may have been a while yet before the liturgy came to be prayed entirely in Coptic. There is no way to know this for sure because the earliest manuscript evidence for any of the three liturgies commonly used by the Coptic Orthodox Church dates from roughly the 3rd-4th century, so we can extrapolate based on that the faith must’ve grown quite a bit in the two centuries after St. Mark’s death, to the point where it was probably completely nativized by the 4th century (around the time of St. Anthony the Great, whose biographer St. Athanasius tells us did not know Greek).
Consider that we have in the above timeline the best case scenario that has ever actually existed in the history of Christianity (the bilingual Greek-Coptic bible fragments are the earliest evidence of a translated Biblical text in the world), and still it took perhaps about 100 years for people to begin nativizing the faith and liturgy, and that process continued on for maybe another 300 years or so after that. (And then there was the whole issue of Arabic after the invasion of the Arab Muslims; it wasn’t until the 11th century that HH Pope Gabriel ordered that the readings in the liturgy be done in Arabic in addition to Coptic). Contrast this with the Bolivia example, where the Church is about 10 years old. In that 10 years it has grown from
one person (the one Egyptian who stayed behind after the few families in the country had left) to over 400 people (none of whom are Egyptians) in the capital city alone, with many more in the countryside. I have spoken with priests and missionaries who have served the Church in Bolivia and they say it is really booming, and the Bolivians really have taken ownership of it and nativized it, and it is very well received into the fabric of Bolivian life. The liturgies are in Spanish, as well as the catechumens and deacons’ classes, etc. It is Bolivian in every way except its priesthood and bishop, and this is again because it is very, very new.
The answer really is that things take time, and it is better that people adapt to Orthodoxy within some kind of known framework, and then have native priests emerge as the population naturally changes (more converts + more nth generation people). There are non-Egyptian priests in the Coptic Orthodox Church in America (though I don’t think we have any convert bishops yet), and as the demographics change, things will only become in the diaspora more and more like they’ve always been in the church at home (where Romans, Persians, Ethiopians, Syriacs, and many others were counted among the monastics of the Egyptian desert, for example). Some churches are better poised for this than others (e.g., the Greeks have been in America for a lot longer than the Ethiopians), but it’s coming to all of them eventually.

've seen recently some videos of mixed English-Amharic-Ge’ez liturgies held by the Ethiopian church in Canada; pretty cool! Friends of mine in Los Angeles tell me that the EOTC they go to already does portions of the liturgy in English, though I haven’t seen it yet.]
Here’s a story you might like about the recent conversion of a German Protestant scholar to the Syriac Orthodox Church:
theorthodoxchurch.info/blog/news/2012/10/protestant-new-testament-scholar-converts-into-the-syriac-orthodox-church/
Trust me: It absolutely does not matter where you come from when you come to Orthodoxy. What matters is that/if you’re here now, and (most importantly) where you’ll be in the hereafter.