Zare, your argument is inherently flawed because it's coming from a collectivist point of view. The problem is not because of the people who support the religion, the problem is the religion itself is in desperate need of reformation, much like other religions have done in the past. I attribute Christianity's path to reformation beginning during the Thirty Years War, when the French, predominately a Catholic based nation, used the political theory of politique to support the Protestant Sweden against the Catholic Holy Roman Empire because they prioritized politics over religion. Gustavus Adolphus made a large amount of gains whilst Sweden were on the offensive, whilst the French supported them. This then transitioned into a French offensive against the HRE when Gustavus died, and eventually, a peace treaty was made. This helped generate religious tolerance between Protestants and Catholics, as well as other Christian branches.
Islam has never had this. Their doctrine is based on imperialism and consolidating for conquered people, which whilst an efficient way to expanding land in times old, is very much a bad idea back then and especially in this time period. Most Muslims, both Shia and Sunni are able to practice their religion in peace, even fundamentalists, but their remain a select few who feel Islam is entitled to the same opportunity that Judaism was given with the birth of modern Israel, except on a wider scale due to the Umayyad conquests, stemming through Northern Africa and into Iberia. This is one of the main reasons that ISIS exists.
It's an incorrect assumption to say that Islam is a religion of peace, because their doctrine proves otherwise. However, in present day, most believers in Islam can practice their religion in peace. Don't place the blame on Muslims, place the blame on the fundamentalist flaws that come with Islam.