Historically, one can look at the Yezidis and see the imprint of Sufism on a substrate of ancient Iranian religious ideas that have long existed in the mountains and appeared in numerous religious movements throughout history. Patricia Crone's The Nativist Prophets of Early Islamic Iran deals with this dimension of the region's history and includes coverage of the Yezidis. Some of the vocabulary in Yezidism parallels that of the Baha'i Faith, founded in Iran in the 19th century. The Sufi dimension was brought in the 12th century by Shaykh Adi b. Musafir, who was in many ways a mainstream Sunni.
Sufism is Islamic mysticism: the quest for direct experience of the divine involving the gradual perfection of the soul. This is a key to understanding the role of pilgrimage in Yezidism. God is unknowable, but created emanations or manifestations of himself so as to be known. Melek Tawus is the highest such emanation, which is why he is sometimes called "God" and at other times referred to as an angel. He is perhaps similar to the yazatas of Zoroastrianism, and I suspect the "ezid" term which lends the Kurdish sect its name is etymologically related.
Now, to Shaykh Adi: like many Sufis, he used strong language to describe his connection with God, and after his death, many people for whom he functioned as a holy man in life came to believe that he was God, having achieved a sort of oneness with the divine essence that is also a form of emanation. Now I'm finally to pilgrimage, for one way Yezidis pursue the journey towards God is through pilgrimage to living holy men or sites associated with deceased ones. Here we get to the intersection with what some regard as simply folk belief, as shrines can be associated with practical benefits like miraculous cures.
One should appreciate, however, that the experience of being Yezidi is tied to the sacred history of the landscape of their homeland. The central pilgrimage, required every year on the Festival of the Assembly in late September, is to Lalish, the sacred space of both Melek Tawus and Shaykh Adi. (At last report ISIS was less than 20 miles from it.) When removed from this sacred landscape, it strikes me that Yezidism itself must inevitably change. I suspect this process has already begun with the growth of a Yezidi diaspora, and Philip Kreyenbroek, whose influential account of Yezidism I have referred to in writing this, has written a book specifically on the faith in Europe.
I am reminded of the Jews in the Roman Empire, especially after the destruction of the Temple that had been the center of Jewish life for a millennium give or take the Babylonian Exile. Judaism changed in those days, with the law gaining elevated importance and the prominence of philosophical speculation with roots in the older wisdom literature. If the Yezidis, too, become primarily a people in exile (despite some communities in Syria and Turkey), then Yezidism, too, will change.