I think it is important to look at where Muslim-majority countries are broadly coming from, with regards to the idea of whether the idea of a Muslim State is valid or not. I cobbled this together from some notes and my memory, so I am sorry if it reads a little disjointedly. On preview - Wow, this got long. Sorry
When thinking about Islam today, exposed as we are to the discourses of modern Islam, it is easy to think of it first and foremost in terms of law. As if the law represents the final point of normative recourse, that all other claims are somehow just claims, but that the law is what Islam is really about. This is not correct.
Major sects in Islam emerged over disputes about what constitutes the legitimate polity. Thinking about where, historically, a link existed between Islam and the polity in terms of secular and religion is unhelpful because from the outset issues of religion and issues of polity were fused.
This does not mean that Islam is a political religion or that there is no separation of religion and state in Islam. Referring to it as such is initially helpful for a basic understanding, but ultimately we ought to leave behind that language altogether, because the societies under discussion did not fit into that western, post-enlightenment discourse.
Islamic lands spread across a tremendous range, and in them what arose were local regimes. These regimes faced the problem, first of ruling, then of ruling successfully. You have to make sure people don’t starve. You have to monopolize violence. You need to have stakeholders who see benefit in supporting you and fighting for you. People have to see that the body politic in which they exist is healthy with plain evidence, so that they are more likely to accept that the ruler is ruling as God wants, in accordance with divine law. The sultan’s legitimacy, therefore, as a divinely sanctioned ruler arose from being a ‘good’ ruler.
This sets up a tension between the state and what divine law looks like. Is the polity that is legitimately administered in accordance with divine law the polity in which the many legal rulings given by jurists are followed to the letter, irrespective of the consequences, or is legitimately Islamic state recognized by its general ends, rather than the means?
Here we see the notion of the Islamic ruler - the one who makes policy - develop in distinction of the legal scholar, who is conducting a private discourse, who has the luxury of not ruling. Their duty is to understand divine law. The rulers made policy. But the Arabs had no empire experience. They had no blueprint to run their empire, so they took what was available: Greek philosophy and Persian statecraft. Rulers came to be seen as physicians throughout Classical Islamic thought, tending to the health of the body politic to create an environment where people can be virtuous and then be happy. Justice, in Islamic societies, is therefore not just about law but also about producing a virtuous and happy society. Of course, what these terms mean and how to achieve them became a debate, and in Muslim political thought, this debate was between philosophical modes of thinking that proceed from reason and experience and scriptural modes of thinking that proceed from the hermeneutical reading of scripture as the foundation of law.
In other words, there is a tension between form and meaning that permeates Islamic discourse on polity throughout the medieval era. Over time, you have sultans who promulgate qanun, a body of law alongside Shariah that derives its legitimacy from the political and divine legitimacy of the sultan. This typically dealt with the person’s relationship with the state (tax, criminal law, etc.) and became increasingly important as modernity was imposed on the Islamic world and problems arose that were not covered by Shari’ah law.
Law, indeed, as the foundation of the nation-state and therefore the modern state has become transformed in the Islamic world. Today, the discourses of law have acquired a definitive primacy in the way that they did not in pre-modern Islam. Law was important but not constitutive. Now everyone is asking, what does a modern Islamic state look like? What do we do with the Muslim-majority states we have now? What Islamic law and how should it be applied? These questions and more are disputed in ways that are new in Islamic history, and the temptation to look back for precedent is almost asking the impossible.
Remember that up until halfway through the 18th century almost no Muslim populations lived under non-Islamic rule. By 1939, colonialism or de facto control by European powers controlled everyone except (secular) Turkey and Saudi Arabia. Modernity, for Muslim lands, is an experience imposed, not an experience that they authored, as it was for the West. It was a deep historical transformation that for many meant colonization, political marginalization, and exploitation. With the destruction and denouncing of pre-modern Islamic political structures as, well, pre-modern, you have a crisis of a simultaneous inferiority complex imposed by the modern narrative of progress, and a sense of grievance because a sense of what was lost and what is currently happening.
Compounding this problem, we also have a class of modern Muslims who are not educated in traditional Islamic schooling but by the state, in a fashion modeled in some way or another on the west, and these people emerge with some authoritative agency. They are not someone who will defer to a traditional scholar in answering the question: ‘what is Islam, is x Islamic or not.’ Indeed, they often see the traditional scholars as the source of the trauma that afflicts Islamic lands. This is the social group that is the foundation of all of the modernist Islamist movements in the Islamic world (as well as almost all of the liberal/progressive movements), and their role with the traditional scholars, and with the West, and with their governments, is very much combative.
So when you are talking about a “Muslim State,” its important to remember that what is going on now is the result of an imposed transformation, with every side proposing something innovative and every side suffering from imposed assumptions. So it isn’t so much that the idea of a Muslim State is invalid, it is that, in a very real way, it is in its modern sense still being reforged and disputed. This is why there is such a problem with trying to generalize from locality to locality, yet we cannot ignore the large trans-local trends and movements happening. It’s important to talk about, because the concept of a Muslim State is something deeply important to millions and millions of Muslims and it matters for the world. But we have to be careful about drawing conclusions about a process that is in such chaos.