very small amounts of evidence for some propositions (e.g. the existence of the deity Thor.)

Very small amounts of evidence? Entire mythologies are quite strong evidence of something thor-like. The point is to be able to say "I don't believe in Thor" and "That is strongish evidence for the existence of Thor" without conflict.

Your point about neglecting small shifts (likelihood ratio 1.0001) is well made, but your numbers are too charitable. When someone says "there is no evidence for X", there is usually some substantial piece of evidence (LR>10) evidence, even quite strong evidence, known to them, not a tiny shift, but not totally conclusive either. The problem is that even substantial evidence usually has the problem you are pointing out (cost of consideration exceeds Value of Inforation).