I prefer Popper's "first principle" to BroJoeK's, in part for the reasons you cite, dearest sister in Christ.
BroJoeK's "first principle" — "natural explanations for natural processes" — rather stacks the deck in favor of finding "natural explanations" only. We tend to find what we are looking for, and screen out anything irrelevant to that purpose.
Thank you so very much for your astute observations, dearest sister in Christ!
Are we really looking to "unstack" the deck, or just struggling for control over who gets to stack it? Is "not allowing a divine foot in the door" any less stacked than allowing only one specific "divine foot" while disallowing any others?
You're almost, but not quite there.
If I could just get you to understand the ancient concept of "branches of knowledge", each with its own realms and rules -- i.e., theology, metaphysics -- then acknowledge that the branch called "natural science" refers to nothing more, or less, than studies for natural explanations of natural processes.
So science is just one branch of knowledge.
If you want other knowledge -- i.e., of the super-natural or ontological -- then you must leave science and go to another branch.
Natural-science by definition cannot deal in those realms.
Of course our problem is that, in the modern-materialistic world, in the minds of many there is no other branch besides "science".
For them, if it's not "science", then it's not real.
But such was never the original intention of thinkers like, for example, St. Thomas Aquinas.
So the problem is not just that "science" has become immeasurably greater than it was in Aquinas' age, but also that all those other branches of knowledge have effectively atrophied to the point of non-existence in our modern minds.
How, exactly, to go about restoring the non-scientific realms I don't have an answer for, however...
Seems to me that was one problem that people like JRR Tolkien and CS Lewis addressed in their marvelous fiction-fantasy works.