Practically, there is a fuzzy threshold beyond which I need to just ignore very weak evidence, unless there's so much of it that the sum total crosses the bounds of significance.

Consider the difficulties of programming something like that:

Ignore evidence. If the accumulated ignored evidence crosses some threshold, process the whole of it.

You see the problem. If the quoted sentence is your preferred modus operandi, you'll have to restrict what you mean by "ignore". You'll still need to file it somewhere, and to evaluate it somehow, just so when the cumulative weight exceeds your given threshold, you'll be able to still update on it.