The best quote I heard about the refugees is "they're fleeing the same problems and people you're worried about". I really hope we can protect refugees better than we have been.
The problem with religion is that it isn't one thing. It's a general philosophy and creation myth to many, and to others it's day-to-day marching orders. That latter set is purely crazy - hearing voices, believing counterfactual things, and is often programmed to kill other groups or at least believe they need to die. If someone is a flat-earther in a battle with round-earthers you can take them up in an airplane high enough to see the curve of the Earth, or to space. But there's no counterproof that can be given to a religious killer.
I think the reason people tend to see Muslims as a single group is that both camps (not Sunni vs Shia, but Sane vs Crazy) share the same sets of religious leadership and respect the same historical figures. For example, a southern baptist and a russian orthodox member would have nothing (religiously) in common. If one sect started murdering others, or picking at a weak justification in the bible for slavery, the other sect would condemn them. But Muslims (through the seemingly accepted method of death-threats from priests) control their religious world more tightly. To be a member is to at least profess to, if not actually believe, a specific set of things.
There's no broad blanket condemnation for entire classes of horrible actions. The Saudis control Mecca and yet the "peaceful" muslims continue to travel there and enrich not only a kingdom but one that routinely executes people for rejecting minor religious traditions. It'd be like me saying I didn't support the Catholic molestation coverup and then going on a three-day Vatican tour and coming back with a pope hat.
So while it may be a stereotype, it is a deserved one (Ask Salman Rushdie and Ayaan Hirsi Ali) and Muslims should pay the price of changing it instead of insisting others simply trust them despite their memberships and alliances.