Actually, the Protestant churches were the ones that tried to set up true theocracies (where the secular government is run by the religion and members of the clergy). Catholicism has always had a very clear distinction between the leader of the secular state and the religious authorities, and has endured much persecution when secular leaders (such as Henry VIII) attempted to take over religious authority.
There have been times when the State and the Church have worked together, and these times have never been good for the Catholic Church, because the State always wins. The Spanish Inquistion, for example, which was originally intended to be an internal disciplinary activity aimed at rooting out tendencies that had crept into the Church in Spain during its Islamic captivity, as well as immoral practices particularly among the clergy, was transformed into a State-sponsored persecution because the Church relied upon the State for certain administrative things and thereby gave itself into the power of the State and its politics.
However, in Christianity, these things are fundamentally separate, and even modern Protestants do not hark back to Cromwell and wish they were members of a state and church like his, which were one and the same. That is because theocracy goes against the nature of Christianity. Even Protestant Christianity, which does not have a temporal head, rejects the idea of a state controlled by religious authorities.
But Islam has been a theocracy since the beginning. It is a syncretist cult that took bits of Jewish ritual law and the concept of the "prophet" from Judaism, some historical figures and the concept of multi-nationality from Christianity, and its black rock at Mecca and demonic concept of a remote, unpredictable god that must be placated at all times by bizarre bloody acts from Arab paganism. Islam requires the imposition of sharia and in fact also rejects the concept of the secular state that it has been forced to accept. I think that unless we understand that Islam wants to impose itself on the totality of human life - because that is its nature, not a corruption of it - we are being fools.
Of course, one look at Islamic societies will tell you how well this model functions! Interestingly, as you point out, it is riddled with strife - but that's because the nature of Islam is such that if it becomes more moderate (from contact with non-Muslim societies, for example), the more orthodox, who actually have the support of Islamic scriptures, rise up and overthrow the moderates. That's what we're seeing right now, but that's what we've seen for 1400 years.
Islam doesn't have a head (although the direct descendants of Mohammed do have primacy), but I still believe it could be destroyed by destroying Mecca because that's where its cult-object is. On the other hand, bombing the Vatican and even killing the Pope would be upsetting, but have no effect upon Catholicism. This is because the Pope is not an object of worship and another Pope would be elected; the main office of the Church could be (and occasionally has been) established anywhere; and while the loss of the symbolic significance of Rome would be sad, it would not destroy either Christianity in general or the Catholic Church in particular. But I think if the Muslims lost their black rock with the hole through the middle and the strange vagina-shaped silver lips around the hole - well, the entire religion would vanish as if its people had been released from a wicked enchantment.