Actually, a “sola scriptura” religion like Islam, especially if the scriptures don’t make any sense, fundamentally rejects reason. Granted, the scripture is in practice interpreted over the years by one or another person officially empowered to do so, but you still have to go back to the fact that the scriptures themselves are violent and anti-rational and cultic.
Islam swamped a number of militarily unprepared societies in the ME and beyond, since the thing that enabled its spread was the fact that there was no major power after the fall of Rome, and there was a division between the Eastern and Western Church that enabled Islam to rise on the ashes of various Arian-inspired heresies. However, there would be a period of a few decades between the time Islam took over a civilization (the modern nation-state wasn’t really in existence at that time, and these were all loosely confederated and generally warring kingdoms) and the time Islamic anti-reason complete extinguished it.
Don’t forget that Maimonides (Rambam) converted to Islam at one point. He may have been forced to do so by the Muslims in Spain. He later went back to Judaism, but he had also written that it was not a sin for Jews to convert to Islam.
To know how anti-reason Islam is, however, all you have to do is look at any Islamic society.
I agree with you that Islam is imbued with inherent violence from its founding. In my opinion, it was essentially the co-opting of religion to serve the political ends of Muhammad, and his main sponsor, Abu Bakr - it was a means to totalitarian dictatorial power. Many cults are twisted to serve the personal desires of their leaders, but Islam has incorporated some of the most extreme enforcement mechanisms of all - death for trying to leave, death for criticizing the Prophet, death, death, death - and unlimited taxation authority, and enslavement and rape, and more death.
I believe that explains why there is a hodgepodge of practices from all the other religions which were popular at that time - Muhammad was pandering to constituencies to gain adherents early on, before he had military power. Jewish dietary practices became Halal, the Jewish direction of prayer was adopted, but turned from Jerusalem to Mecca. The Christian liturgy of the hours was supplanted by five daily prayer times for muslims, and Christian fasting during Lent became Islam’s Ramadan. Pagan pilgrimage to the Kabaa in Mecca, became the Muslim Hajj, and the Supreme Diety of the Pagan pantheon - the moon god Allah - became the Muslim supreme diety.
In addition to the historical conditions that you mentioned which assisted the rise of Islam, both of the Great neighboring empires, Byzantium and Persia, had recently had their cities ravaged by plague, throwing them into economic crisis and leaving no one to police the wild hinterlands of Arabia. Additionally, a gold discovery in Arabia gave the Arabs a surge of resources.
Although, I agree with you that Islam is fundamentally flawed from its founding, there is enough in its Sola Scriptura, the Quran, (and the rest of its tradition) from other great and ancient traditions, to extract good. In fact, it has already been done many times in the history of Islam, and in the lives of many muslims. Even today there are many sects of Islam which are pretty peaceful, but they are being overwhelmed by the lavishly funded Wahabbis and Salafis, since the Arab oil boom.
How it is interpreted is key. As Benedict pointed out, there are elements of our own scriptures which could be problematic, depending on their interpretation. In some ways Islam is like a counter-revolution back to the old testament in its graphic warfare and conquest. We have come to terms with violent elements in our scripture by containing them in context to their time and place rather than holding them up as eternal and universal. And importantly, we have applied a moral filter and used judgement to hold moral ideals as a paramount test.
I object particularly to two principles used by extremists in interpreting the Quran and Sharia:
1. The principle of Abrogation, which establishes that any chronologically later statement supersedes any earlier statement, when there is a conflict (there are many in the Quran). Well putting aside that ANY conflict shows that Muhammad was not a reliable prophet for an all-knowing God, picking the later verses (when Muhammad was engaged in escalating campaigns of conquest) skews everything to wartime footing. It appears to me that Muhammad only became more megalomaniacal as time went on, until he was finally poisoned, making abrogation a fundamentally dangerous standard to use. Better a standard of what is meant for eternity (enduring principles of morality) vs. what was meant for those particular circumstances (which will never be exactly replicated).
2. The principle that Muhammad is the best possible example of human conduct. This establishes that whatever Muhammad did is good, and that no one can have the standing to say otherwise, as he is the best. I believe that this principle was established to defend Islam against attacks which would fundamentally undermine it - that Muhammad was clearly not a true prophet, based on his bad conduct. The perverse result of this principle is that morality and reason must be overridden to accept murder, rape, slavery, theft, pedophilia, etc. Morality and reason are completely subordinated by this principle of interpretation.