Some pentecostal / charismatic / holiness traditions lean in the direction you describe, as do Science of Mind adherents ("Christian Science" ). In fairness, many evangelicals would totally agree with the last sentence of your post, and even the whackier fundamentalists usually stop short of things like letting their children die for lack of medical care.

Still, taken to its "logical" conclusion, if god is willing to miraculously intervene in human affairs, and his representatives claim the promise that "the prayer of faith shall heal the sick", then the hospitals should be empty by now and most theists would have converted to these "name it and claim it" types of denominations.

I always tell these kinds of folks to go empty out a children's cancer ward and be sure to invite the local media to record the proceedings. For some reason they never take me up on that.