Expected value matters. If the cost of being right vs wrong is heavily skewed, a slight (or in your words, 'weak') change in evidence supporting a conclusion may have a huge impact on the expected value of the outcome.

Think of it like a gun that has a 99% chance of being a harmless toy. If it only has a 1% chance of being a real gun, does that mean that it is okay to play with it like a toy?