To demonstrate the error of your position, we need look no further than the Sunni/Shiite split within Islam, and we can go further and show schisms within Sunni Islam and Shiite Islam. Islam is not unified and there is no essentialized or idealized Islam. The Muslim world reflects the many different ways Islam is understood, interpreted, lived, and practiced.

You talk about a “religion grounded in books — all written by different authors at different times, and compiled over decades or even hundreds of years, you are going to have contradictions and ambiguities which leave ample room for different interpretations.” I assume you believe this to be true of Islam. It is not. While the Hadith may have many stories about the prophet and his life from many authors, the Qu’ran is written by just one man. The prophet wrote the Qu’ran himself and designed it to make himself the leader and the perfect human being. (I’d say that gets things off to a rocky start to begin with eh?) While the religion began like any other, it was not catching on. He spent 13 years trying to get people to buy into his new religion with very little success. With only about 150 followers, he decided to leave Mecca and goto Medina. Along the way he was short on resources and decided to start raiding and robbing other travelers. Then he had a brain storm! He made this part of the religion! He began to get people to follow him to share in the spoils. Once the religion involved profit from piracy and murder and robbing, it began to get quite profitable and popular. ¾ of the Qu’ran had been written and it was fairly peaceful But once the Prophet became a blood thirsty warrior, the texts took a huge turn and became violent. Mohamed understood that there was a conflict with that so he made it clear that the later verses abrogated all the earlier ones. So every single thing that was written that favored non muslims was replaced by texts which condemned them. Two thirds of the Qu’ran is devoted to non believers. That is political not religious. There is nothing good said about non believers and no tolerance is encouraged not in the later texts. But it is hard to see that because the earlier texts were mixed with the old. The Qu’ran was later bound to have the longest chapter be the first and second longest the second and so on. This makes it incredibly confusing and I suppose that was the intention. Once established in Medina he made Sharia law very clear that people were to be converted or killed. The Muslim calendar does not begin with the Prophet’s birth or the beginning revelations of Islam, it begins with him establishing an army. From the get go Islam has been a political force. It is still the duty of every Muslim to work tward enforcing Sharia on the entire world. The world is divided into tw parts, Dar Islam and Dar al harb. House of Islam and House of War. House of Islam includes all Islamic countries and those who have been subjugated and acknowledge Islamic Supremacy and those who do not and Islam is therefor at war with. Islam is at war with the West. And no matter how many times our leaders tell us that we are not at war with Islam, Islam is at war with us.
I’m sorry but there is plenty of reason to be afraid. Not so much of Islam but of what our “leaders” are refusing to do to protect our freedoms. This is what you should look at. Islam does not have many heads. It is very clear. It has many different degrees of followers but the goal is the same for all: Islamic domination. Some achieve this with violence, some with words, some are lazy but they all have the same goal – a goal we should be concerned about.