Trevor,
As you can see from the replies so far, "It ain't no big deal."
My sense when I first read your post is that you're worrying way too much about this issue. Learning to clean a sensor when necessary is easy. Learning the "best choice" of cleaning materials for your particular camera/sensor is easy to research on various web board dedicated to your camera type. Ultimately, with almost any mirrorless system today what you'll find is that 99.9% of dust spots you get will be quite easily remedied with using a good quality puff blower (rocket blower, etc.). If you were to get something more stubborn, learning to do a wet cleaning isn't nearly as dangerous, traumatic or mysterious as it might appear, unless you have real problems with fine muscle motor control. Any average, fairly handy, person can do it with relative ease.
In fact (though I'd not necessarily recommend it) I just read a thread on another forum where lots of professional photographers have resorted to just using good quality Q-tips to clean their sensors!
So, don't worry about it. Both Fuji and Olympus "dust shakers" are quite effective. Go for the camera that fits you best! Learning to keep your sensor clean is both do-able and a normal part of owning an interchangeable lens camera system.
Rand