Most major spiritual belief systems include at least one kind of 'third group' of beings that exists besides humans and divine entities, usually spiritual in nature, (the Hindu Naga, the Islamic Jinn, various east asian fox spirits, the greek nymphs and other spirits, various new world entities that range from little people and to hairy wild-men). In many places, mostly rural, belief in these entities (in their various forms) are still generally accepted and are often mentioned in actual religious texts; meanwhile, belief in elves and fairies are regulated to, at best, New Age belief systems in most European countries. Did these beliefs die down naturally, or were there any major religious aspects to the separation of 'superstition' from 'religious' beliefs in terms of the belief in the existence of non-human sentient beings?
Religious orthodoxy is effectively impossible in a premodern state. The religious experiences of people in medieval Europe, and indeed elsewhere, as you have stated, were very personal, and the exact form of Christianity practiced could change from village to village. Many places were still awash with local tradition predating their Christianizations, much to the ire of the bureaucratic clergy. One example of such a role is the Hungarian taltos. They were essentially the leftovers of shamans from the days of pagan Hungary, but of course conformed to the "skin" of Christianity, as it were. One Taltos, an Erzsebet Balazsi was accused of being a witch, to which she defended herself, saying that, "the Taltos cures, finds buried treasure, and fights for Hungary in heaven*. They drew the attention of the Hungarian priest Arnold Ipolyi, who wrote of them as being a very famous Hungarian folk tradition. Another such example of an isolated tribe maintaining their "pagan" trappings were the Sarakatsani of Greece. Patrick Leigh Fermor, who was perhaps the most hellenized Barbarian in history, wrote as recently as the 1930s that they were Orthodox, but did not know of the Holy Spirit, combining the son and father into a deity called Ays, and that they paid great reverence to nature spirits and to the spirits of animals. So your answer for when people put aside the fairies was at least more recently than 1930.