The Church was not established in Rome but rather in Jerusalem. Greek was actually the lingua franca of the Church for most of its first three centuries. The Roman Emperor Constantine made Christianity the official state religion of the Western Roman Empire and along with the Emperor of the Eastern Roman Empire signed the Edict of Milan which granted religious tolerance across the Empire in 313 AD.
So, both Latin and Greek were in use. Latin was more prevalent in the West and Greek in the East. The Western Empire was staggered by barbarian invasions - Huns, Visigoths, Ostrogoths, Vandals (from whence we get the term vandal), Angles, Saxons, Jutes and Franks. Rome withdrew its legions from Britain around 410 AD because Rome itself had been vandalized - literally.
With the collapse of the Western Empire you see the beginnings of the development of feudalism. Everyone was far more concerned about their survival than Rome. Over in Ireland (where Rome never conquered), Irish monasteries kept alive writing (in Latin) and Roman “civilzation” while Britain was being sacked by Angles, Saxons, and Jutes. The Irish monasteries were able to send out missionaries into Scotland and from Scotland down into Britain and eventually into Gaul (France). Latin was the language.
Sorry about the long history lesson, but it is important. The Eastern Empire survived until the Turks (Ottomans) captured Byzantium (Constantinople/Istanbul) in 1453 AD. Greek was always the primary language not Latin.
Latin is significant in traditional Catholicism because it was the language used by our ancestors - our western ancestors that is (looks around for Bishop Basil

), I am a calligrapher and illuminator and one of the most heartbreakingly beautiful books (The Book of Kells (the four Gospels) was written in Latin. There are oodles and oodles of manuscripts all in Latin. Latin was the language of the educated throughout the Renaissance and a surprising amount of English words are rooted in Latin courtesy of Shakespeare and his contemporaries. I have not even touched upon Gregorian chant, etc.
I grew up with the traditional Latin Mass and was an altar boy. I never had any problems with Latin and it is dear to my heart not to mention the fact that cognates leapt off the page and proved so helpful in my study of Spanish. Can I speak Latin? No. Anymore than I can speak Spanish but I can read Spanish fluently.
Latin connects me with my ancestors many of whom suffered greatly for their Catholic faith. Tradition means a lot to me as does knowing that if I had a time machine and could drop back into the Renaissance, I would be able to attend and participate in Mass.
Latin was a unifying force in the Church.