That seems very commonsensical and should be very uncontroversial.

I knew this in the abstract, but wasn't adhering to it properly in practice. See my remarks about the shift in my beliefs about Penrose's views on consciousness.

As a general rule, we should not expect there to be strong pieces of evidence in favor of a false position.

But there are often apparently strong pieces of evidence in favor of a false position. That's the point of the latter half of the "Major weaknesses of the “single relatively strong argument” approach section of my post.

In practice, it's often the case that all we have is weak evidence — the situation is just that some evidence is weaker than other evidence. It can be easy to deceive oneself into thinking that the relatively strong evidence is stronger than it is.

However, there always will be a lot of weak evidence (at various values of 'weak') for or against anything, it's just too easy to come up with using motivated cognition.

I agree if you're talking about "sufficiently weak" evidence. But consider the example of quantitative careers and earnings in my post. I believe that the individual arguments supporting it are genuinely weak, but that there are fewer arguments of the same strength against the claim, so that there's not much of a risk of motivated cognition skewing the conclusion. Do you disagree?