There is logic in your argument, but the logic stops the second you start applying bigotted interpretations of the facts based on very primitive and fundamentalist views of religion.
Yes, Mohammed was involved in a holy war. But this means many different things to many different muslims.
For some it is a spiritual and internal war, for each person to realize their ultimate goodness. For others, holy war and Jihad implies an act of self-defence -- not an aggressive act. Even now, to fight against someone who is persecuting you is something that is universally held as a noble reason to fight. In 700 AD, that might also involve decapitating the enemy. These are equally valid as the interpretation you go for, that holy war is a war against anyone not Muslim because they are not Muslim. There is no logic in fundamentalist interpretations of a religious text, because most religious texts can even seem contradictory at times -- you only need to pick the quotes that support your view. In other words, your argument is only pseudo-logical.
Yes, the Koran does criticize non-believers. But you take the most fundamentalist interpretation, once again. To most Muslims, non-believers refers to anyone who actively attacks and persecutes the muslim faith -- in 700 AD these threats were polytheists. And, in fact, some parts of the Koran praises Jews and Christians -- they inspired Mohammed, after all. It's only the fundamentalists who interpret "non-believers" to refer to anyone not muslim. In fact, fundamentalists go a step further and persecute fellow muslims who don't interpret the Koran the way they do. This is not the "accepted" interpretation, it is the ignorant interpretation.
Of course people want to "follow the example of mohammed", as you said. But again, it's in your interpretation. Jesus lived in a house without electricity. Is a man who lives in a house without electricy more pious than a man who lives in a house with a computer, telephone, and television? Some people think so. But for other people, they put the actions in context. The way you follow Jesus has nothing to do with the kind of house he lived in. Likewise, many muslims do not applaud the acts of fundamentalists who would kill strictly because someone is a non-muslim. This is not the example Mohammed meant to set -- even if Mohammed happened to kill people who were non-muslim.
Allow me to repeat: there is no logic in fundamentalism. What you are talking about -- the interpretations of the religious texts -- is a fundamentalist interpretation. To say that to be a good muslim is to copy everything Mohammed did without paying attention to differences between the 20th century and the 8th century is a fundamentalist interpretation. You seem to be the only person -- aside from anti-muslim hate groups and fundamentalists themselves -- who believes your argument is "pure deductive logic" without any kind of interpretation and assumption about the Koran.
Implementing this warped and narrow view of all religions would NOT make Civ more realistic, it would make it LESS realistic. Also, it would take away from strategy to be forced down grossly simplified paths like "Islam is the war religion" if you "pick" your Civilization's religion. So not only is your view of Islam -- and all religions -- ignorant. But it also makes for a lousy game of Civ.