The problem is I don't know how to show the absence of arguments for central banking, so a quote by someone as authoritative as Mankiw is really useful, even though it certainly isn't logical proof.

I think it is definitely too presumptuous. First of all you're asserting at least one property of economists that you haven't even tested (uncuriousness) rather than a property of the economics literature which you've at least started to search (doesn't have any well-known theoretical proofs that central banks are a good idea). Also your second argument is at least 2 experiments removed from having a leg to stand on.

What is the difference exactly between the economists and their literature? I mean, if they should be asking a question, and I can't find the answer to the question in the literature and I can't even find the question itself anywhere, and I can't even find anyone pointing out how weird it is that the question isn't asked, what should I conclude other than a lack of curiosity?

Economists are pretty comfortable asserting self-interest as an explanation for anything where it possibly could be an explanation in the absence of disconfirming evidence, because we're that confident in the generality and power of self-interest as a motivator. You would need to do the experiment proving that it isn't self-interest--if the subject were anything other than central banking.