I would say that proclaiming something inexplicable and beyond the realm of human intellect is antithetical to the entire concept of investigation. Jedweber's link is a perfect example or how such denominations treat such mysteries - by declaring them outside the bounds of human study and reason. From their standpoint science can't possibly contribute because theological mystery "exceeds the natural knowing power of the creature", "lies above the finite intelligence", "results from the weakness of the human intellect, which, like the eye that gazes on the sun, is blinded by the fulness of light." For them the solution itself is the mystery. It's not something that's meant to be solved because it already is. There is nothing left to study; the fingerprints, the DNA, the body, the smoking gun, the signed confession, the written testimonies of over a dozen eye-witnesses, etc. are laid bare before you, but your puny human mind is just too weak to put it all together and see the big picture. They view supernatural truth as being the ultimate optical illusion - there's nothing to be decoded or studied or better understood, we simply fail to accurately perceive it.
Their concept of mystery here is fundamentally different from ours. Investigation of any other mystery would involve the pursuit of evidence with the expectation of greater understanding. Investigation of "theological mystery", however, involves presupposing the truth of specific religious tenets and accepting marked areas as being completely off-limits to understanding. Contrary to being open to science for clues they have already declared this mystery unsolvable - case closed, end of story, pencils down. Such behavior doesn't foster inquiry, it shuts it down. In the end the only thing that sets them apart from the fundies is that apparently whenever the magic decoder rings start getting too close to anything which could potentially be tested or investigated they start going haywire and shooting off sparks. I can certainly understand the appeal of a religion that works to make sure its every assertion is entirely unfalsifiable, claims to be an ally of science with the buzzword of "compatible", and boasts about being the golden mean between fundamentalism and angry atheism, but it's the way they rationalize such a situation to themselves that I find difficult to wrap my head around. It's basically divine revelation with static, declaring their pipeline to God to be strong enough to glean certain absolute truths which are beyond them, but not others.