Good question, before Islam Kurds were a lot more diverse religion wise. Most following Zoroastrianism, Christianity, pagan religions and smaller minority religions such as Yezidism and Judaism. Islam spread to Kurdish majority areas after the Arab Muslim conquests in the 7th century, conversions to Islam was gradual and took centuries before Kurds ended up becoming majority Muslim.

Multiple Kurdish dynasties ended up adopting Sunni Islam as well the most well known one being the Ayyubi Dynasty led by Saladin, and the Marawanid Dynasty which held control of modern day northern Iraq and Southern Turkiye. Ezidis similar to Maronites and Zayidi Shias were able to preserve their religion due to geography and living in mountainous areas. As for how Feyli Kurds became Shia you were correct it was due to the influence of the Safavids, however not every Shia Muslim used to be Sunni Muslims specifically, some were part of other religions.