When we find a law with no lawgiver, we call it God’s law.
When we find order without an organizer, we call it creation.
When we find a justice or fittingness in the affairs of men beyond their capacity to engineer, we call it providence.
It is remarkable how naturally our mind recognizes these cases of intelligibility-beyond-intelligence and how difficult it is to devise a metaphysics in which they are at home.
It would not do to identify God as a person from whom creatures and laws are entirely separate. God’s order for the operation of beings is intrinsic to them, and His laws of righteous conduct are intrinsic to justice, but in each case they seem to transcend their subjects. At least, this is what our tendency to attribute them to God would suggest.
Nor when we speak of God’s laws, of God’s order of the world of nature and of men, do we just mean that these things appear as if imprinted by divine intelligence. It may be that these are nothing but appearances, but to say so is not to explain what we mean when we speak of them, but to deny it.
It is remarkable how unsatisfied people are with scientific explanations. They ask how something with the marks of intelligibility came to be. Give them an answer deriving it from some general principle–spontaneous symmetry breaking, detailed balance, natural selection, or whateer–and they will be disappointed. “Oh, so you’re saying it’s an accident.” In other words, if that’s the explanation, then there is no explanation. Scientists are proud of general principles that explain many phenomena, but what people are looking for is a single cause that directly imprinted this particular intelligibility.
Remarkable as well that we monotheists at least attribute all of these cases of intelligibility-without-intelligence to the same beyond-intelligence. One could say that honesty demands that I tell the truth, sobriety demands I avoid drunkenness, patriotism demands that I avoid treason, et cetera. To say that God demands all of these things is to bring them into relation with each other, that there is something inconsistent about recognizing the authority of some of these demands without recognizing the others. Similarly, the intuition of God’s created order might be one reason we so often speak of “nature” instead of “natures”, even though if we were good Aristotelians we would definitely prefer the plural.
The historical ubiquity of belief in divinity is a scandal to philosophers, perhaps especially to a believing philosopher. He does not imagine that the reasons his uneducated co-religionists could give for their beliefs would pass muster with his colleagues. And yet, he does not want to credit their faith to fortuitous error or over-hastiness. How can the common man of all ages have gotten so deep into metaphysics so easily? How could he have gotten so much right with such invalid reasoning?
There may well be an inverse relation between the logical strength of arguments for God’s existence and their attraction to the religious mind. I can’t imagine that any non-believer is troubled by an argument from morality (of the “no law without a Lawgiver” sort); arguments of this sort seem quite weak to me. But it does that the virtue of addressing God in a way that believers actually conceive and relate to Him, and so no less figures than Newman and Lewis understood its power.
I think the common intuitions are valid, although by themselves they hardly constitute proof, are hardly even clear enough to know what they might be proof of. I doubt we ever cease to rely on them, at least as a check on our reasoning.
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Throne and Altar 
I think you would really, really enjoy a book by a British theologian of the 20th century: A Science of God? by Austin Farrer. (Sold in the US under the dreary title God is Not Dead.) He says what you are saying in an more extended way. Farrer was Lewis’s good friend and confessor. He remains my Master, even though he was Anglican and I am Catholic.
I suppose the reason my atheism stubbornly refuses to heal is that the world does not seem so intelligible to me at all. We just keep adding epicycles to our models, they become more and more predictive, but logic is in the model, not in the world. The world isn’t logical in itself, it is just that we are able to construct logical models that approximate it better and better, but logic is not an inherent part of the world, it is the screws and bolts our models as machinery are built of.
I am a software developer, amongst others. I have never ever seen anything that was not man-made that lent itself well to the logical structures of code. And even human behavior is not very logical to us, humans, let alone things outside the human sphere. If you want to model a bunch of animals with code, there is no feature in nature that tells you from which angle to model them, whether to do the cladistic angle biologists actually use, where the Dolphin class inherits from the Mammal class, or the morphological angle, where the Dolphin class inherits from the AquaticPredator class. Is-a-mammal, behaves-like-fish.
What successes conservatives had so far came from stubborn anti-intellectualism, refusing to see the world from the perspective of any grand theory, and just sticking to whatever experience taught us to work. This never grants one academic respectability, but does the job…
There is a parallel intuition that the intelligibility of the world, while accessible to us, is not not proportioned to us, is not like the sort of thing that comes out of our minds. For example, one could argue about whether the Schroedinger or Heisenberg formulation of quantum mechanics is in some sense better, but clearly it is only human minds that make a choice, rather than nature itself. The theory is the same whichever mathematically equivalent way one chooses to look at it.
I’ve heard speculations that nature is in some sense like code, but I think it’s always the sort involving zillions of cellular automata from which the macroscopic world emerges, the sort of explanation that makes ordinary people say “So, you’re saying it’s just an accident?”