I suspect a part of the appeal of this saying comes from a mental unease with conflicting evidence.

I am quite sure you're onto something here. A similar effect occurs when people try to argue that a given intervention has no downsides at all; none at all? Really? It will be absolutely free and have beneficial effects on everyone in the world? Why aren't we doing it already then?

People aren't used to thinking in terms of cost-benefit analysis, where you say "Yes, it has downsides A, B, C; but it also has upsides W, X, Y, Z, and on balance it's a good idea." They think that merely by admitting that the downsides exist you have given up the game. (Politics is the mind-killer?)