Many scholars see that it was a combination of Greek and Christian thinking that was responsible for the Enlightenment. Greek thinking by itself was incapable of initiating science as we understand it.
Indeed Greek thinking is incapable of initiating modern science, which is why the Enlightenment (and before it the Scientific Revolution) had to discard Greek philosophy in order to move forward. The synthesis of Greek philosophy and Christian theology as represented by scholastics like Thomas Aquinas got in the way of the development of modern science. Galileo was persecuted as much because his findings conflicted with Greek philosophy as with scripture.
There is one Christian theologian whose thinking did genuinely contribute to the modern age, and that is person William of Occam who adopted a position not dissimilar from that of the Muslim theologians.
William of Occam insisted that reason and revelation operated in two
completely different spheres. The method by which we know things depends on nature of the object to be understood. The nature of God, human morality,
and the metaphysical world could be known only by means of revelation
whereas the physical world could best be understood by inductive reasoning
which involved experience and experimentation. No room was left in this
paradigm for most of Greek philosophy, which relied on abstract ideals and
depreciated the concrete, but room was created for the emergence of what
would later become the modern scientific method. This would not happen until
the sixteenth and seventh century voyages provided the economic means for a scientific revolution, but I would argue that the theoretical basis for one was laid out two centuries earlier.
Your story of logic and reason is a child of the Enlightenment.
Understand that the 'logic and reason' of the Enlightenment is not logic and reason of the Greek philosophers. It is inductive, not deductive and conceived in terms of 'common sense' as Thomas Paine used the phrase.
Its founding 'gospel' was that science and education would eliminate poverty, hunger, war and disease. How's it going on its promises?
And that is where the Enlightenment makes its fatal error, in my opinion.
The Enlightenment represented an attempt to apply the scientific method to all areas of human life including those, which Occam had relegated to religion, such as the nature of God and human morality. It saw no function for revelation whatsoever, since all knowledge could be deduced through the study of natural law. Initially this led them to deism, a conception of God in terms of an impersonal essence much as the Greeks conceived but eventually some philosphers like David Hume argued that ultimately it was impossible to prove philosophically that there was any God whatsoever and one can certainly not deduce morality from nature.
