The narrative that you have put forth is the narrative that is found in many textbooks, pop history works, documentaries, and other fields of "common knowledge". This narrative is also wrong quite simply.

Now, there is some truth that the language talents of Western Europe to translate new copies of Ancient Greek texts were greatly diminished in the former Roman territory of western Europe. However it is hardly the case that all of Western Europe was sundered from Greek/Classical learning as a whole in the Middle Ages. Indeed it is frankly absurd to think that there was only a resurgence of contact between the West and the East following the sack of Constantinople in the 16th century, especially when Latin had been ruling former Byzantine and Arab lands in the Eastern Mediterranean for centuries. For example, following the 4th crusade the Greek and Balkan heartlands of the Byzantine empire were administered by Latin dynasties, and much of the Levant had been either invaded or occupied by Latin Christians off and on for centuries.

How did manuscripts and so on get transmitted in the classical world and the Middle Ages?

Well obviously the first barrier is that language barrier. Many works of the classical world were composed in Greek, and knowledge of Greek did decline in Western Europe, even if it did not vanish entirely. Most people would speak either Germanic or Romance vernacular languages, and formalized language education was the domain of the Church largely. However there were other languages in use in Antiquity, or at least that ancient texts were transmitted into. Classic authors often employed Latin writing as well, and later Islamic texts, written in Arabic also were available for translation, particularly as the 11th century rolls around and texts from Arabic writers such as Ibn Sina, known in the west under a Latinized name, Avicenna. Indeed the transmission of many Arabic written texts that translated classical works were not newly arrived in Europe during the 14/1500s, but rather in the 11th century! This is for example when new Latin translations of Galen's works and Aristotle's books start to get re-circulated in western academic circles. Indeed it is now the case that for some works, Greek and Arabic translations of classical works have now since disappeared and their Latin translations are the only extant versions. Given the propensity for Medieval monastics to create new copies of texts, this is actually how many ancient texts survive today. Without Medieval monks toiling away at preserving the works of classical antiquity, our surviving Latin corpus would be much much smaller. However this was not the only route that texts from Classical Antiquity took to arrive in the Middle Ages, as some just never left! The medical texts of Dioscordes for example were still in circulation in Western Europe up through the Middle Ages, and there are even vernacular translations of his works into English!

In short, some texts never left the west, having already been translated into Latin which never disappeared as a language of learning, others were reintroduced through Arabic or Greek versions, but often, much earlier than the Renaissance.

Now the other part of your question that I want to address is this

And what role did the church play, if any, in suppressing the works of ancient Greek philosophers and historians?

None. Indeed it was the Church who was responsible for preserving the majority of what has survived to today, and certainly what was in circulation back in the Middle Ages themselves. Many people today hold that the Church of the Middle Ages was an oppressive and regressive institutions that burned out any trace of non-conforming prior learning in its zeal to create a thoroughly Christianized society. This is not really the reality of the situation and has more to do with Reformation era stereotypes and popular misunderstanding of the Middle Ages (as well as a certain strain of anti-Catholicism often seen in the Anglophone world). Throughout the Middle Ages the great intellectual work of the western wold was happening under the auspices of the Catholic Church.

This took many forms. This could be Medieval monks creating copies of classical texts, or developments in philosophy and so on from Church members. The Medieval Church owed an intellectual debt to giants of Classical learning. Figures such as Augustine of Hippo for example were towering intellectuals of their age who produced truly prodigious amounts of philosophy, theology, and other writings. I may be mistaken on this point, and I am happy to be corrected, but Augustine may be the single most prolific author of Antiquity who works have survived to the modern day, largely because of his importance to the Church. However the Church was not merely preserving the works of its own. Later on in the Middle Ages, as Latin translations of older works circulated more broadly, new intellectual developments took their ideas and ran with them.

There are many figures of Medieval philosophy, theology, science, medicine, history, and other fields to contend with. But the figure who stands head and shoulders above the rest is St. Thomas Aquinas. To say that the man was a prodigious writer would be to sell him short, to say that he had a mastery of Greek, Latin, Arabic, Biblical, and Jewish sources would likewise only be an understatement. As a part of his output he did things like

-harmonize Catholic dogma with Aristotelian philosophy and write extensive commentaries on Aristotle's surviving works

-incorporate translations of Islamic and Jewish writers into his work

-wrote over 8 million words in his articles, books, defenses, and so on. (remember, Augustine is far and away our largest surviving Latin writer and Aquinas has him beat by a solid 3 million)

For his efforts, he was canonized as a saint a few decades after his death, despite some controversy over some of his writings, he has remained the corner stone of Catholic philosophy ever since.