This reminds me of the distinction between complicated and complex systems.  A complicated system can be predictably modeled, even if it requires a large model or many steps to do so.  A complex system, on the other hand, is inherently unpredictable and surprising.

Mathematics and other axiomatic systems are very complicated, but still fundamentally predictable.  Arguably physics is complicated rather than complex (though with the observer effect in quantum mechanics, I'm not totally sure about that).  The weird thing is that, even though, say, biology is "just" physics applied to living organisms, it is incredibly complex, unpredictable, and surprising.

Your point about humans being able to shift between systems is particularly salient.  If reality is fundamentally ordered and knowable, we would expect a coherence between various axiomatic systems insofar as they reflect reality, even though no one system has the whole picture.  So we have to remain humble and recognize the extent of our ability to know on our own.

All that said, if we develop and test various axiomatic systems to show they have some bearing on reality, then we ought to be able to triangulate some knowledge about the fundamental reality that those systems describe.  Catholicism recognizes this as "natural revelation," by which, with the aid of reason, we can observe the created world to come to know some things about the Creator, who is ultimately outside of any systems we can devise.

The medieval theologians had this idea of "divine simplicity," which says that God is identical with all of His attributes.  So we might say "God is omnipotent," but in reality, God is not a being who possesses omnipotence, he *is* omnipotence.  And also omnipresence, omniscience, goodness, love, and so on.  Thus God, who is the fundament of reality, is fundamentally simple; He is one thing.  Everything we can say about God is just another way of triangulating on that ultimately simple Being.

Which brings me to a question, based on the end of your post: 

If you are appreciative of and wistful towards religion, what is keeping you from making the plunge and pursuing it further? 

 I'm confident in my compass. This is distinct from  being confident in my beliefs. 
Chemotaxis, or how cells move up a chemical gradient for nutrition, or phototaxis moving towards or away from the light gradient. They don't "know" what is the maximal position to be in, but they move in the direction that is strongest to  help them survive. I argue that this is not entirely mechanistic, that there is a sort of 'taxis of meaning' where agents move towards the highest signal that is meaningful towards them.

That eternal 'search' to something is what I see in Complexity, coupled with being entangled  with the environment. I see all humans doing that in some sense, but what is it? Something arch-typical in mythology and religion is the concept of 'the greatest good'. Every religion has it, I personally don't see myself aligning myself with one completely at every time.

Like this taxis, I use my compass to help me find that which is most meaningful to me, trying to absorb the traits of 'the greatest good' that I find in the world. This can change moment to moment and I'm okay with that. 

I love the lecture series on religion by Jordan Peterson, Eric Weinstein's passion and references to his  Jewish heritage even though he is not religious, Alan Watt's lectures on the Abrahamic Religions, Hinduism and Buddhism. All these are individuals that embody that trait which I want to also take on. But the spiritual practices that I align most with would likely be Taoism, Buddhism,  and particularly the perspectives given by John Vervake and Jordan Hall regarding "The Religion that isn't a Religion".