TL; DR: The shift described here never took place. Greek religions and philosophies have always had strong elements of both the mystical and the rational (which are not necessarily mutually exclusive).

The assumptions implied here are very large. The question implictly assumes that the philosophers so widely revered now were the norm for their time, and reflective of a culture which was founded in rationalism and reason. This is not necessarily the case - Socrates was famously sentenced to death by the vote of popular assembly for allegedly "corrupting the youth of the city" with his philosophies and teachings (as described by Plato in 'Phaedo'). Even when he was still alive, he was openly mocked in Aristophanes' comedy 'Clouds' - although being featured in a comedy is certainly not necessarily indicative that he was universally hated or disrespected, it does indicate that he and his followers were considered laughable/mockable by enough people for it to appeal the the masses.

Even then, the philosophers that we revere as bastions of reason were not removed from the mystical aspects of their religion at all. Again in 'Phaedo', Plato goes to special efforts in order to portray Socrates as a religious and pious man. In Athens (where most of the philosophers we associate with Ancient Greece came from), the Eleusinian mysteries were an enormous part of the culture, and Plato/philosopher gang seems to be no exception - Plato writes, 'the mysteries have very real meaning: he who has been purified and initiated shall dwell with the gods'. Mystery cults/religions and focus on the mystical were integral to Greek religious life and thought - even Phythagoras, who we think of as a mathmetician and by extention rational, was the founder of his own Phythagorean school of mystical thought, closely related to Orphism.

Moving onto Christian theology, it too has always been a mixture of the rational and the mystical. Debates on topics such as the nature of Christ's divinity are so wrought with technicalities, debate and reasoning that single solutions on doctrine have not been created for over a thousand years. On the other hand, appreciation for the mysterious nature of God is considered by a many a central part of being a Christian.

It might seem paradoxical, but I hope that has awnsered your question in some shape. I personally find that the presense of the rational and the mystical make both these things all the more interesting and beautiful, but others don't, I suppose it's a matter of taste.