The modern day diversity argument is just that modern as we like all things pluralistic.
The Byzantine rite are those that adhere to the liturgical services of the imperial church. These were the greek speakers and thus looked towards the capital city for clergy, support and direction during and after the schisms and various arab conquests. Alexandria for instance became a tiny fledgling community of greeks, that looked towards the capital to be sustained, in the 10th century a large bedouin tribe converted to the byzantine church when a section of that region was captured by the byzantines. This obviously required orthodox evangelism of which was supplied by the imperial church of whom were only familiar with the DL of St John Chrysostom.
Most in the Antiochan Church were muslims whom converted back to christianity in the 10th century after the byzantines recaptured (arab) lands (969-1085). Obviously it was missionaries from Constantinople that re-evangelized them and supplied priests to this area, etc.
Regardless of what anyone says this is for the best. Diversity is a modern construct which is detrimental to the Church. Its precisely because all of Orthodoxy uses the same so called 'byzantine rite' that there is an Antiochan Church today. After the unia apostacy of 1724 the few remaining Orthodox christians were allowed to appeal to Constantinople to give them a bishop. It was under greek bishops from Mt Athos (who only knew the byzantine rite) that saved the Church in Antioch. Slowly many returned back to Orthodoxy and today the Antiochans are the largest christian church in Syria.
Also the byzantine rite encompasses many services and 'rites'. The Liturgy of St Basil the Great is celebrated 10 times a year. The Liturgy of the Pre-Sanctified gifts was written by Pope Gregory the Great. The Liturgy of St James is celebrated once a year in Jerusalem and on the island of Kerkyra from time imemorable. Any Synod can decide to celebrate the Liturgy of St James and St Mark on their feast days. Then theres the various services celebrated through out the liturgical cycle. It is this Byzantine Liturgical cycle alone that has preserved the fullness of the catholic faith. It is the very vehicle which passes down from one generation to the next the complete deposit of faith.