This is one of those threads that requires Middle Eastern specialists and experts in political Islam.
In broad strokes, it seems to me like the Cold War political history of the Middle East was first dominated by secular, transnational Arab Nationalism, which was defeated by a combination of American skullduggery, Saudi money funneling to religious groups, and the failure of the Arab Nationalists to make good on their grandiose plans. Stuff like Baathism in Iraq and Syria aside, the rest of the region were more or less generic secular autocratic governments, up until recently where a huge swath of Mesopotamia has gone all Taliban-shaped.
There are different stripes of Islamic parties, from the Christian democratic moderate kinds, to more conservative groups that strike fear into the heart of the West even though they might not be very violent (and different traditions within that, from Qutbism as practiced by the Muslim Brotherhood to Salafi? parties), and finally the modern Muslim Khmer Rouge transnationalist caliphate reenactors of the Islamic State, Boko Haram, etc.
Long story short: what if recreating a caliphate was not a violent fringe extremist movement? What if it was something more like Arab Nationalism, uniting the Middle East based on Islam?
I think it's probably impossible, because there's like dozens of Islams, and I'm not even talking about the Sunni/Shia split. There are different schools of jurisprudence. There are other types of philosophies within Sunni Islam. Plus, at the end of the day, each region, country, and ethnicity has Muslims that practice it differently from the other guys beyond the hill, and they're not going to agree upon on one standard version in order to create a religious Muslim version of the European Union. How are they even going to choose one caliph?