If I understood correctly historically there had been a continuity of at least one theocratic Caliphate in the Muslim world. The Umayyads replaced the Rashiduns, the Abbasids replaced the Umayyads and the Ottomans claimed supreme religious authority from the Abbasid Caliph when Egypt was conquered in 1517. But when Turkey abolished the Caliphate, why was there no replacement? Why didn't say Saudi Arabia,which controlled the Holy Cities after the Ottomans vanished, or any other major Islamic powers, set up a successor Caliphate?
The Arab, Turkik, and Persian world had become too diffuse and diverse to accept a new Caliph, or continue on with the extant Caliphate of the Ottomans, if there had been calls for a revival.
Hussein, in Western Arabia, and Abdulaziz, both desired hegemony over the Arab World (Leatherdale, 1983). In some respects, as the Arabs had allied with, and had the support of, the victorious Allied Western powers, they became the preeminent group within the Islamic World (Abdullah Mohammad Sind, 2013), and were arguably in the best position to form a new Caliphate. The division between these two men, the machinations of the peace settlements of WWI, including the Sykes-Picot agreement and the Balfour Delclaration, meant that true Arab unity would never be achieved. Even though Abdulaziz' capture of Mecca in 1925 meant that Saudi Arabia now controlled the most holy of sites, division still existed.
As for Turkey, Ataturk's reforms were meant to both modernise and save what remained of Turkish holdings from encroaching European powers (Asyraf Hj Ab Rahman, 2015). As a result, Islam and its relation with government was reformed for a more progressive and modern state to be ushered in. The conflict and division over religion's role within Turkish politics continues to this day. However, Turkey would not hold enough sway to realistically establish a Caliphate.
I won't go into my condemnation/commentary on the so called-ISIL. This isn't the forum for it. Suffice to say, they're seemingly very evil.