So if I found an amateur painting that was centuries old in an attic somewhere, and posted a scan of it on my blog, I would then be the copyright holder of that painting? That feels so wrong.
No, you would be the copyright holder of your photograph.
My understanding of copyright rulings in the US (eg see Bridgeman Art Library v. Corel Corp. above) is that a faithful photograph of a painting is not considered "original". Therefore, if I do not have the copyright to the painting, neither would I have copyright over the photo.
[1] https://en.wikipedia.org/wiki/Autobiography_of_Mark_Twain#Ma...