The problem you are encountering are the equivocations of naturalism and supernaturalism.
Naturalism, most cleanly, is the embrace of a set of tools to establish what is and isn't in our world. The tools are empiricism, when possible the scientific method (which is a formalization and refinement of empiricism), and reasoning. This is "methodological" naturalism, and it is a definition and approach that falls in the realm of epistemology.
There are epistemological alternatives to naturalism. The most notable is the use of intuitions to establish reality. Many philosophers' resort to philosophic intuitions to argue a point, and intuition is used in many other fields as well. Intuition is explicitly non-rational, but may sometimes be "empirical", although not scientific. Our neural net processing is non-rational, but is often based on empirical inputs. For those who hold by "direct realism" rather than "indirect realism" -- we can have direct apprehension of what "is" and this direct apprehension is therefore an explicit intuitive "knowing". Philosophers from across the ontological spectrum have asserted direct knowledge as the way to establish their ontology -- from Ayn Rand asserting materialist direct realism, thru Gautama Buddha asserting the unreality of all dualities thru direct apprehension.
In this use of "natural", the opposite, Supernatural, would be knowledge we can have, but which we can't acquire naturally. The directly intuited knowledge is supernaturally acquired, as it is not thru methodological naturalism.
"Naturalism" however, is often equated with "materialism". Or with "physicalism". Or less commonly, with a commitment to scientific pluralism. In this usage, Naturalism is a combined claim -- one uses methodological naturalism to do epistemology, BUT there is an ontology embedded into the methodology. Many, plausibly most, "naturalists" are asserting this sort of dual claim, ontological as well as epistemological.
The dual claim meshes well with scientific reductionism, and leads in a fairly straightforward manner to scientism -- the claim that science, or science-like activities, are the only way to gain valid knowledge of our world. Advocates of scientism claim that naturalism requires scientism as a conclusion.
Under the ontological naturalism view, anything which is non physical, or non material, is "supernatural". This word is then applied primarily to Gods and spirits, but also can readily be applied to all abstractions, most notably morality. Many scientism and/or reductive physicalist advocates dismiss morality as unreal.
There are a lot of problems with the absolute, scientistic, ontological naturalism POV. The one that philosophers of science have pushed back on hardest is the metaphysics you reference. Science does not self-justify. The justification of science, or of empiricsm, and the development of the frameworks of thought within philosophy, CANNOT BE SCIENCE. Therefore, not all knowledge can be scientific knowledge. And theorizing on thinking, and how to do epistemology, sure doesn't look to be material, or physical. Applying methodological naturalism to absolutist scientism as a theory, it is tested and refuted.
Note this breaking of the lock of "science" on knowledge in relation to philosophers, breaks it as well in relation to other fields of knowledge. Art, history, etc -- all the humanities and fine arts have an equal claim to provide knowledge as philosophy does. The critique of the too strong claims of scientism -- leave epistemological naturalism in some disarray.
The breakdown of reductionism within science has been another problem for ontological naturalism. If the other sciences don't reduce to physics, and starting in the 1990s, the consensus of philosophers of science was that they CANNOT (See SEP on Scientific Reductionism, section 5), then "its all physics" seems to be false too. Most physicalism today holds that the other sciences, and the phenomena they study, are REAL, and do not reduce, but are somehow DEPENDENT on physics. The nature of this dependence is still TBD, and basically is awaiting the development of emergence as a firm theory and field of study.
You also pushed on another problem -- we appear to directly apprehend logic, math, and morality. This would put all three in the epistemological "supernatural" as well as the ontological "supernatural. But logic and math are PART of ontological naturalism, and essential to doing science, so the ontological naturalists seem to be sabotaging epistemological naturalism.
The response of many physicalist theoreticians to the problems that logic, math, pluralism and metaphysics offer, is to accept the reality of the non-physical world of abstractions, and assume that abstractions are also encompassable by an even stronger theory of emergence (where non-physical things can also emerge from physics). This is also the dominant view of consciousness today among physicalist philosophers of mind -- consciousness is non-physical, but is emergent from and dependent on physics.
This theory of very effective emergence, to remain "physicalist" requires that emergent phenomenon cannot be independently causal -- IE ultimately all causation is thru physics. This is a bit of a problem for math, logic, and consciousness, as consequences in all three areas appear to flow directly from prior events or conditions, and none reduce to physics (per the reason that emergence theory was proposed)!
There are also more epistemological problems for even this emergent physicalism, in that people can reason about, and perform tests upon, God and spirit hypotheses. Every consistency or historical accuracy test for a "sacred" scripture is such a test. As is The Problem of Evil. And the Thomist "5 Proofs of God" clearly show that we can reason about God. So just as math and logic are allowed into the expanded "emergent non-reductive physicalist" model, so plausibly are God and Spirit claims. If they are epistemologically naturally evaluable, and "ontological naturalism" is letting all sorts of other phenomena into its tent, then why not Gods and Spirits? Also, if one is an ontological naturalist, and emergence theory is TBD, why assert prior to its development, that it MUST NOT include causal independence for emergent phenomenon?
These two points lead directly to Hempel's Dilemma. If one is committed to epistemological naturalism, then what is or isn't revealed in our investigation of the universe, cannot be pre-constrained. To do that pre-constraining, is to violate the methodology of naturalism, and the basic principles of science. This leads to Hempel's Dilemma. Paraphrasing "IF one defines physicalism so as to exclude the things that physicalists want to exclude (independently causal consciousness and abstract objects, and God and spirits), then that definition can be shown to be false. There is no definition of physicalism that is not either false, or allows the opposed concepts to be part of physicalism".
The tangles that ontological naturalism gets into, and the clear incoherence of its use of "supernatural", strongly suggest that the useful philosophic approach to both terms should be to use them only epistemologically.
Naturalism in a purely epistemological sense then would be to use methodological naturalism to acquire knowledge. Supernaturalism would be to assert there are methods of knowledge acquisition that use direct intuition, and/or are non-rational. Supernaturalism might then play a key role in justifying non-materialist/non-physicalist ontologies, but otologic questions would be a downstream issue, not embedded into epistemology.