There are some so universally accepted misconceptions in history that many people have difficulty in even understanding that they can be challenged at all. One of these false ideas is that the Catholic Church was the only vehicle that could have saved the best ideas of the Roman civilization for us.
One part of the same claim is that, without the Catholic Church, the western civilization could even have disappeared completely. This is simply a guess over one possible outcome, but it has been spread as a fact for centuries. However, the facts on the ground do not support this educated guess.

Most of all, Christians themselves did forbid and destroy all of the Epicurean, Stoic and other philosophical schools in the Empire of Rome. If they would not have done so, these schools would have quite probably been accepted and flourished also under the new Germanic rulers of the Western Europe.
Quite probably this would have happened in a similar way as Roman Christianity was allowed to continue its operation also under Germanic rule. The victorious Germanic tribes had lived under the influence of Roman culture for centuries. Many of them wanted nothing more than to adopt the Roman customs and traditions.

However, at the time when these Germanic tribes achieved a position of power in the areas of modern Italy, France, and Spain these areas had already been utterly and forcibly Christianized. All remnants of the older Roman religious and philosophical traditions had been erased from these areas by the time of the German invasion.
One other complication was that most of the Germanic conquerors had by then already adopted Christianity. However, they had acquired it in its ‘heretical’ form of Arianism. This fact did lead them to needless confrontations with the newly Christianized Western Roman Empire. It was also one of the decisive factors that did lead to its demise.

Many these tribes had been allies of the Roman Empire for centuries. However, adaptation of the ‘heretical’ Arianism made them all too often to be seen as enemies of the newfangled Christianized Roman Empire.
The needs of the uniform Christian state church or the future Catholic Church dictated that they were often treated as enemies. This did happen in a situation where Rome would have needed all the allies that it could get.
These facts have avoided and tiptoed around by most Christian historians, as accepting and publishing these facts would not have benefited their pet ideology.

The Roman empire under Hadrian (ruled 117–138) showing the location of the Roman legions deployed in AD 125. - Wikipedia

If Christianity would not have achieved its position of a totalitarian state church in the Empire of Rome, the real wisdom of the ancient Greeks and Romans would quite probably have been transmitted directly to new generations though the many philosophical schools that did exist before the Roman Christians suppressed them.
For example, the Epicurean, Stoic and Platonist schools were dutifully erased form all cities of the empire by the extremely intolerant Christians.

Catholic Church did originally suppress most of the legacy of the ancient Greece and Rome. Only with the advent of Renaissance, and the weakening of the position of the church, was the legacy of Greece was allowed to surface again.
Up to that time the Catholic Church allowed only those Greek or Roman ideas to be studied that did be seen to support their own new and strange ideas.
In practice, just Aristotle and Plato were among the allowed philosophers. Even their work was, in fact, quite totally suppressed for centuries. In the end, they just were the only Greek philosophers who could be misused to at some level to support the new Christian notions over gods, humans, and the universe.
Of course, I am only guessing here. However, so are those who do say that without Catholic Church there would not have been no other forms of education in Western Europe in the ‘Dark Ages’.

There is even compelling reasons to suggest that there would not have been ‘Dark Ages’ if there would not have been the rise of the intolerant and extremely narrow-minded Catholic Church. Nobody actually knows what would have happened if the new Germanic rulers would have acquired the traditional Roman religions and also the ideas of traditional Roman religious tolerance.
We do not know, what could have happened if they would have acquired also the wide mix of religions and philosophical schools and also the ideas that were present in Rome before the advent of the intolerant Christian religious totalitarianism.

The monolithic Catholic Church did occupy the monopoly of education for centuries for its own purposes in the Western Europe. However, nobody can predict what kind of education there could have been if the totalitarian Catholic Church would not have acquired its position of sole allowed ideology.
Nobody can predict what could have happened if the Christians would not have been able to destroy all competing religions and the old schools of philosophy from the Empire of Rome just before the final Germanic onslaught.
However, it is similarly hard to understand why Christians feel free to guess that without the spread of their narrow-minded ideology, the older and much more open-minded Greek and Roman ideas would not have survived. In real life, the main enemy many of these ideas was just the intolerant Christian ideology.