My question is less so about actual Christian and Greek thought and more to do with the education system in classical to late antiquity and into the early middle ages especially in the Roman East.
In classical antiquity, it is my understanding that Greek philosophy was the foundation of education in the Roman world as Romans with means would educate their children using "the classics" such as Homer. The wealthiest of Romans would even have Greek philosophers as house slaves to directly educate themselves or their children or to be advisors for politics and policy. Everybody wanted to emulate Alexander the Great and have their very own Aristotle so-to-speak.
As Christianity spread within the Roman world and the network of authoritative ecclesiastic bishops began establish legitimacy, there seems to be a big shift in how ancient people's sought education going to their bishops for instruction rather than traditional Greek choices. Many of the New Testament books are essentially written as authoritative instruction to specific communities for example.
And in classical to late antiquity, Christians like Origen of Alexandria would write extensive works directly debating contemporary "pagan" Greek writers like Clesus. So there seems to be a struggle for the "soul of scholarship" in this period. However, I think it should be pointed out that folks like Origen were likely Greek themselves or at least very well educated in the Greek system. The Wiki page on Origen for example has some info about his early life and upbringing.
So, to condense my question into a nutshell: Is early Christian thought more like a completely independent strain of scholastics, or did it more-so evolve naturally out of the Greek education system and was more like a continuation of it? I hesitate to say the latter because Greek philosophy was vastly more decentralized than the organized and ordained Christian thinkers of this time period. Please correct me if the narrative I outlined above has inaccuracies too. Thanks!
Christian education and scholarship of Late Antiquity was very much a process that grew out of the pagan educational/scholastic system that had arisen in the late Empire. Ecclesiastical figures such as bishops, and saints, often grew up in the late Imperial educational system, and it produced some of the finest minds of Antiquity in St. Augustine, Gregroy Nazianzen, and Basil of Caesera among others. This was trying for some Christian students, as at this time the educational systems of the late Empire, around the early 4th century, were thoroughly pagan, however this does not seem to have stunted or severely impacted their ability to obtain the education and benefit from it.
It was only over the course of the 4th century, as Christianity seeped into the broader culture at large that this started to shift. Christianity sat in an uneasy relationship with the often still resolutely pagan religious landscape of the Empire, and it was only over time that Christianity got the institutional power to start radically altering the nature of scholastic life in the empire. There was a, brief, effort to kick Christians out of the educational life of the Empire under Julian the apostate who tried to restrict Christians to only works written by Christians, but due to his brief reign his efforts to roll back Christian influence were unsuccessful. As the 4th century wore on and the Christian elite came to dominate the Empire there were essentially two competing systems.
The first of these systems was the traditional educational system of the Empire that emphasized learning and education in the "classical" model, namely the study of rhetoric, grammar, philosophy (Mostly Neo-Platonic philosophy), and classical works of literature such as Homer. This was how much of the empire's bureaucracy and institutions were staffed, by the products of these systems of education and social networking. (Even back in the 4th century, the people you met and shoulders you rubbed in education was almost as important as the content that you learned). Christians actively participated in and perpetuated this educational system. Indeed as the Church institutionalized over the course of the 4th century the Church, not just the Imperial administration/court, started to rely on this same system to produce administrators and functionaries who could oversee the Church's burgeoning economic and political power.
The other system that arose in contrast/opposition to this system of education and social connection was the rise of ascetic life of monastic communities, especially in Roman Egypt and Syria. These communities were a hotbed of elite activity and interest, despite their vows of poverty and remote, often harsh, lifestyles in the desert. Nonetheless, some elites of the empire still flocked to the establishment of new monastic communities and became influential through their religious affiliations. This is how figures like Gregory Nazianzen became influential and powerful in the Christian community (in addition to his family's wealth and importance)
Over the centuries that followed in Western Europe the Imperial system of education collapsed as the Roman empire's ability to maintain its institutions failed, and over time it was the monastic/church path kept Roman works of literature and philosophy alive in the west and eventually gave rise to a very similar system to the imperial education once the Middle Ages rolled around and cities and states started to grow again.