There already is a thread on this paper but I’m happy to re-open if you want to discuss further.
On prayer: As I take you to you point out in your last paragraph, science can rule out scientific explanations on the efficacy of prayer or lack thereof. It cannot rule out a supernatural intelligent agent who chooses which prayers to grant and decides to thwart such experiments. Of course, one could argue theologically about whether that is in the nature of the God one believes in.
I agree that the label “pseudoscience” is incorrect for such supernatural explanations, with the proviso that supernatural explanations characterized as being science do qualify for this label.
I think there is a commonality with the genealogical approach to Adam: that approach is an explanation of how a belief could be true, and be compatible with science. But it is not a scientific explanation.
That’s probably enough for this thread.