Ah, but it's not really a solution at all, merely a tool to facilitate investigation. The difference is the difference between methodological naturalism and philosophical naturalism. The philosophical naturalist says "The material is all that there is." The methodological naturalist says "We will behave as though the material is all there is."
Science is about investigating the material world, and for that, methodological naturalism - the assumption that material events have material causes - is an absolute prerequisite for science, because science cannot proceed into the realm of the non-material. That's not what it's for, and not what it's about - science is about the material world, and must necessarily restrict itself to investigating the material world thus. It does not declare that the material is all that there is or all that there must be, it only says "the material is all we will investigate".
This is my point, there may be a point where the natural sciences should simply respond "we do not know."
But never a point where one should stop looking - you may have to settle for "we do not know...yet". ;)
Here is the true nature of the debate. It is not a question of faith versus reason but of epistemology. I will agree with you that on the research level a naturalistic assumption is necessary. But it is when the theories are presented in the classroom as fact and not as a working hypothesis that we get into trouble.