The mistake, in my opinion is not acknowledging that the problem IS Islam in its entirety.
There are currently 1.6 billion Muslims in the world. Do I think every last one of them is out to kill the infidel with every waking breath? Of course not.
I am convinced, however, that Islam is not just a religion. It has no separation of church and state by its very nature, and it is the very definition of a totalitarian ideology.
How do I know this? I've done the reading.
When people compare Islam to Christianity (disclaimer, not a Christian), the only real way that comparison is even remotely sound is to compare something in its current form (Islam) to something as it existed centuries ago.
Even then, a pre-Reformation, height-of-the-Inquisition Christianity was a very different philosophy than Islam has always been and continues to be.
At the core, the teachings of Christ are very very different than the teachings of Muhammad. Source? Again, I've done the reading.
Muhammad carried a sword, personally led armies into battle, and one of the central tenants of Islam, and you can deny this all you want, is to spread that Ummah until it encompasses the whole globe.
What Jesus had to say was far, far more benign, and the proper comparison to make between Jesus and another religious 'prophet' is to compare his philosophy to that of Buddha.
Buddha and Jesus would have got along fabulously.
Muhammad would have tried to convert both of them to Islam at the tip of a sword.
As another point, Hinduism is also a quite benign theology in comparison to Islam.
I'm sorry, the history of Islam is predatory, expansionist, and almost always violent.
The only reason everybody seems to have forgotten that, is that with the gradual embrace of science by Christendom over the last few hundred years, the mostly-Christian West has been so wildly successful and advanced so far beyond the rest of the world technologically, that after the fall of the Ottoman Empire one hundred years ago, yes, Islam, for the first time since it's creation, didn't pose a material threat to everyone else.
Now, the West has shared their science with everybody, the playing field is again much more level, and once again, Islam has the means to try and spread that pesky Ummah, which again, is one of the sole doctrines of the ideology that is Islam.
Once again, Source? Did the reading.
Re-read my first post in this thread. Learn about the Crusades. Learn about Vienna, Tours, the Barbary Pirates, etc, etc, etc, etc.
I'm not making this shit up. I wish I were. I really do.
TL:DR You are the one making the serious mistake.
p.s. Most estimates (did the reading) put the percentage of Muslims that we would consider 'radical' at about 20-30%.
That's 300-500 million 'radicals', and if you think that sounds preposterous, you're not paying attention to everything that's going on in the world.
Most estimates (DTR) put the percentage of 'radical' Christians world-wide at a fraction of 1%. There are 2.2 billion Christians, so you're still talking about 10-20 million potentially dangerous Christians, just to be fair. There are indeed nutjobs to be found everywhere. Islam just has far, far more than everybody else.