An interesting question. Let me offer a different angle. You don't have weak evidence. You have data. The difference is that "evidence" implies a particular hypothesis that the data is evidence for or against. One problem with being in love with Bayes is that the very important step of generating hypotheses is underappreciated. Notably, if you don't have the right hypothesis in the set of hypotheses that you are considering, all the data and/or evidence in the world is not going to help you. To give a medical example, if you are trying to figure out what causes ulcers and you are looking at whether evidence points at diet, stress, or genetic predisposition, well, you are likely to find lots of weak evidence (and people actually did). Unfortunately, ulcers turned out to be an bacterial disease and all that evidence, actually, meant nothing. Another problem with weak evidence is that "weak" can be defined as evidence that doesn't move you away from your prior. And if you don't move away from your prior, well, nothing much changed, has it?

Overconfidence is a huge problem. Knowing that you don't understand how the world works is important. To the extend that people believe that they can learn significant things from history, "weak evidence" can often produce problems. If you look at the Western Ukraine policy they didn't make a treaty to accept Russian annexion of the Krim in return for stability in the rest of Ukraine. That might have prevented the mess we have at the moment. In general political decisions in cases like this should be made by doing scenario planning. It on thing to say that Britian and France should have declared war on Germany earlier. It quite another thing to argue that the West should take military action against Russia.