> How can an education system teach evaluating sources, and acceptable methods of evaluating facts, when those two things are at the heart of almost every disagreement in the areas we're talking about?

You can disagree about what the evidence is, or what it means, despite having properly followed the principles of reasonable inquiry. The issue is more identifying when those principles haven't been followed. The problem is that school doesn't really get into the tug-of-war between different strands of thought, it's quite parochial: you are told there are galaxies, not about the great debate that led to us knowing that.