Science indeed doesn't answer all meaningful questions. However, your last comment suggests to me that you are expecting or hoping that naturalistic science will ultimately be able to come up with a Grand Theory of Everything, achieved purely through the use of reason and ingenuity. Supernaturalism does indeed posit a great mystery, but if that's the reality, then we have to live with that. Naturalistic science would then have to accept that it is somewhat limited in what it can actually achieve, instead of making hubristic claims about its possibilities [scientism]. If we at least accepted that there is a great mystery, then we could begin to tentatively explore it – this is what is commonly called esoteric, or occult science.

I'm not sure what you mean by science and naturalism supporting themselves. That is at best merely an opinion, presumably the one held by dedicated naturalists. And is it even true?

What is 'science' here? We have to distinguish between true science, i.e. impossible-to-dispute scientific fact, provisional science, i.e. the currently best understandings which may need later revisions, philosophical beliefs posing as science, and statements made by professional scientists.

A good example would be ESP – telepathy, remote viewing, clairvoyance. According to 'scientific' [atheistic] naturalism, this is impossible. We therefore get firm denials from people like Richard Dawkins, even though ESP has been proved scientifically beyond reasonable doubt – see the writings of Dean Radin, especially The Conscious Universe [which includes a great chapter on the psychology of ESP deniers].

Here we have an example of a 'scientist' talking bullshit. So is what he says 'science'? Why is he so intent on denying ESP? Because in his mind it suggests supernaturalism, the denial that the brain generates consciousness, which he believes passionately without any evidence [that's why it's the Hard Problem], which is not great 'science'.

So back to our starting point. ESP is evidence for some kind of supernaturalism. So science should start to explore that seriously, instead of denying it for philosophical reasons.

And I'm sure you're aware that when the early quantum physicists, some very great scientists, began to understand the true nature of matter, they started making statements which sounded like the ancient Eastern religions – Sir Arthur Eddington, Sir James Jeans, Max Planck, Erwin Schrödinger. Fritjof Capra and David Bohm from the next generation took on the mantle and continued on the same theme.