Hmm. You seem to be thinking of deductive reasoning, with straightforward rules, and no judgement in applying them. Empiricism is not like that. Empiricism does not work with "simple thresholds", but with judgement calls. What constitutes evidence, the strength of a piece of evidence, etc., all are judgements. When there is a claim of evidence for something new, the peers of the claimant examine the evidence themselves, consider alternate explanations, and make predictions about what ELSE we could reasonably suspect if the claim were true, then look for that too. Support for a claim is a cumulation of evidences, and the claim is a reasonable inference to the best explanation. Note that most of what we postulate in physics, we do not actually directly observe. Energy, entropy, quarks, etc. -- all are not DIRECTLY observed, they are all unobserved inferences to best explanation.
For ALL empiricism, there is always the possibility that our inference is wrong. We can use probabilities of a prime number series NOT being random by comparing to observations of other stars, and the signal patters we see from them, which appear to be mostly random white noise. But white noise can produce patterns on occasion by chance, so that is always a possibility. That is why SETI does not just look for one thing, but a variety of possible patterns, and the possibility of then interpreting other signal patterns, and extracting more possible information out of background noise. If THAT works, and we find there are messages PLUS the primes, then the "random vs communication" debate would end up settled pretty quickly.
SETI provides a good reference for thinking about trying to evaluate evidence for a "supernatural" designer. To infer design, one first has to postulate a character, and intentions, to an agent. Then look for possible consequences from those inferences. AND the design hypothesis has to be the "best" explanation, in that it knits more evidence in a coherent story, AND makes useful and testable predictions.
I also suggest you drop the use of "supernatural" here. The two most common usages "about Gods or spirits" and "not within the realm of evaluation thru empiricism or reasoning" are often equivocated. You want to DO "methodological naturalism" on a spirit hypothesis, and calling the hypothesis "supernatural" would put it outside that possibility per the second definition. Better terms would be "discarnate agency" or "spiritual designer", because they would help people plus yourself to avoid the equivocation fallacy while thinking about the hypothesis.
Back to evidences. Note I said we need to make assumptions about a designer. IF one postulates the Biblical God, then there are a LOT of "characters" and "intentions" one would expect based on claims and descriptions in that reference about that God, and LOTS of test cases one can look for, in addition to possible traces of design in the universe. What has happened for most scientists, is that there are enough failed predictions form those inferences, that the vast majority of scientists consider the Biblical God to not be a useful or best explanation for our universe -- and for that hypothesis to be effectively refuted.
However, this is not the end of the story. ALL hypotheses are infinitely klugable, with the possibility of special case assumptions added to explain away apparently contrary evidence. The scientists who DO believe in a Biblical God, have to make a lot of special-purpose assumptions to counter the falsified predictions from the Bible. For the literalists, this induces reasons to reject lots of scientific consensus in multiple fields. For the non-literalists, this includes accepting that the Bible is reinterpretable in very non-intuitive ways -- such that its character descriptions are not useful for making testable predictions. This makes the Biblical God NOT the "best explanation" for most testable features of our universe.
The issues that a Biblical God hypothesis encounters, are even more informative than the SETI experiment in evaluating discarnate designer hypotheses. If one postulates a designer of our universe, one must postulate SOME kind of purpose: Life, Beauty, Uniqueness, or SOMETHING, in order to have an intention to test for and provide predictive explanations. We can test our universe for uniqueness -- and the commonality of materials, of galactic structure, and of star structure, repeated billions or quadrillions of times, is NOT consistent with a uniqueness intention. Same with beauty. There are SOME aspects of our universe that are beautiful, but many more which are not.
Life seems to be the most useful of design intentions, and that was the one widely held, until the process of evolution was discovered, and that provided a better "best explanation" for the structures and effectiveness of living things. Design would predict life to appear fully formed, and optimized -- but life appeared late in our universe, and at first very simple. And while life tends to be optimized, it is a peculiar optimization with major design flaws embedded -- like the human lack of vitamin C production, and our hanging our organs off our spine not our rib cage, which make a lot more sense evolutionarily. Our ancestors lived off a diet of fruits, and did not need the vitamin C self-production that most critters have, so losing the ability to make our own vitamin C was not a problem for them. And organs hung off a spine is fine for the four legged creatures that we evolved from. There are multitudes of these features of life that are far better explained by a gradual evolutionary process, where later optimizations have to adapt around architecture choices made for different environments for ancestor species. A pure design process should in contrast have everything optimized without these sorts of poor architecture features, and the ability to explain these non-optimizations strongly supports evolution over design.
A universe designed FOR life is still a possibility. Our understanding of physics is that the constants of the Standard Model are contingent aspects of our universe -- they could have been almost anything. BUT -- when we run what-if physics models, almost the entire range of possible values for most of them would prohibit a universe with life. The universe would exist only for an instant, or blow apart and have effectively zero density and no interactions, or all the mass would end up in black holes, etc. So -- if the possible range of constants is what we think it is, the odds of a universe being able to support life is astronomically small. This thesis is called the Fine Tuning hypothesis. and while astronomers and physicists initially resisted it, it has now been mostly accepted as a reasonable interpretation of the Standard Model. However, an alternative to Fine Tuning has been identified, and that is Apparent Fine Tuning, in which multitudes of universes are created (the multiverse hypothesis), and all life would then find itself in one of the rare "Apparently Fine Tuned" universes, thru an Anthropic Principle.
There is currently no evidence for multiverses, so if a Deist life-intending creator hypothesis could produce some other useful predictions, that could be a possibly more useful prediction than a multiverse. Deism is not strong on predictions, so that aspect of a design/creation model is hard to make useful. The potentially more productive area to look for predictive power is thru looking for other ways that spirit/discarnates might interact with our world. The work of the parapsychology association to look for psi phenomenon https://parapsych.org/articles/36/55/what_is_the_stateoftheevidence.aspx, combined with the failure of physicalism to answer the hard problem of consciousness, provide some support for the possible plausibility of a Deist design origin to our universe.