Then you shouldn’t have a problem expecting science to fill the gaps in our understanding of observed phenomena as it always has in the past. Religious myth isn’t always dogma fabricated out of whole cloth; often it is based on observations that x happens when we do y. Even if these observations aren’t made under rigorous conditions, the element of recognizing correlations between x and y is a practice of science, not religion. The religious element is in the mythical explanation; without knowledge of germ theory, the religious might conclude that washing your hands before eating is a way to stave off a curse from God. They’re not wrong that it works, they’re wrong about how it works. Germ theory corrects them. Science has always been the way to determine how things work the way they do. That’s why in matters not yet apprehended by science, it is not unreasonable to expect that we might one day find a scientific explanation. It is unreasonable to expect to find a religious explanation, because religion doesn’t have a separate method of investigation. Anything we actually find, we find through science.
The only gaps we can’t reasonably expect science to fill are the ones fundamentally uninvestigable to it, and the only advantage religion has there is a lack of rules and accountability. It can be adjusted to fit any paradigm or it can be held dogmatically in the face of alternatives. The problem with this is that once you accept a religious explanation for a mystery of the universe, you risk importing all kinds of religious baggage along with it that has a real influence on your decision making. I don’t think I have to name any examples to impress on you how catastrophic this can be. If your religious explanation does not carry any baggage with it, on the other hand, what does it matter? It’s just a conversation stopper, same as calling it a brute fact.
There is no functional difference in such issues as you describe. A curse alleviated by washing hands, or an antiseptic procedure to stave off illness, is practically the same.
A secular ritual of hand washing prohibits a secular curse of infection, while in a religious context, a ritual stops a curse that manifests as infection. In fact, this can be argued to be in support of the religious belief, confirming it in this hypothetical, as some Jewish scholars have argued that the increase of Taenia Solium in pork and mercury and allergy in shellfish, confirms Kosher laws.
So really, this is not a particularly good example, and only argues for the method of understanding of ritual. Either way, ritual would be paramount, whether done in accord with a secular belief structure or a religious paradigm. It actually doesn't address the religious claim that a divinity is ultimately responsible for the working of the world, who would preside as it were, over the pathogens, and told his followers how to overcome them. Germ theory corrected nothing, but previous theories of how illness was spread here. It did not invalidate any curse in your hypothetical. If we can show that the tides are moved by gravitational pull, that doesn't mean that the set-up of existence was not done in that way by a divine hand.
This is still trying to force animal husbandry and mathematics together. They are about different things, as I tried to explain with pagan sacrifices. Roman Catholics believe in transubstantiation of the Host into the actual body and blood of Christ, but scientific investigation would still just show bread and wine. Does this matter? Not really, for that is not the point. Religion is about understanding and acting within, a metaphysical framework of reality - Science simply trying to understand a practical, material one. Religious-based claims that are just folklore or superstitition, most would accept a scientific 'debunking', but religion is not about describing our world, but interacting with it on a higher plane, as it were. So ritual or mythopoeic usage, need not reflect 'brute facts', nor would they matter there. As I explained with the Egyptians, there can be multiple explanation for phenomena within one religion or even within one thought, but as they don't invalidate each other, neither would a materialist one do so. The only reason someone would decide it does, is if they start with a axiomatic acceptance of materialist explanations as paramount for some reason, and therefore diminish other ones. That is merely a question of one's philosophy, not 'facts' as it were. What are facts? How are they determined? Why does one set of determinations predominate another?
This is not a problem religion has ever had, as it is not about describing our material world, I reiterate. This is why the Bible has two creation narratives in Genesis, or two separate genealogies for Jesus, etc. Do you think the scholars who pored over those texts, and redacted them sometimes, didn't notice?
They are very different intellectual endeavours, which is why appeals to empiricism or 'lack of evidence' are such silly things. You have to preach to the choir for such things to be effective.
Either way you have to assume a worldview, and work from within it. Religion answers question from within a religious paradigm, which is not arbitrary, but stepwise, in spite of atheist ideas to the contrary. A Trinitarian cannot deny the Incarnation, nor a Muslim Mohammed. Great Intelectual systems were created by Religions, by very intelligent men, and these aren't just rationalisations of inconvenient facts or ideas, in spite of the vain modern attempts to write them off as such. This is alongside Scientific Method usually, as seldom, if ever, do they contradict. They only do so, if the Metaphysical imported beliefs somehow gives precedence to certain interpretations, be it religious or scientific. Neither by nature MUST be true, nor need even be contradictory, except by your own assumed axiomatic values. It is merely another Metaphysical System, an assumed Materialism, placed against another system that recognises things beyond that as well. It is essentially Plato's Cave all over again.
This is where Materialism cannot confirm materialism by its own rules. Any such attempt is a Petitio Principii, for any appeal to material reality assumes the primacy thereof. This means the relativity of the value structure is laid bare, and this is why Materialism tends to Nihilism. Nothing can be established if Materialism cannot establish itself. A line in the sand needs to be drawn somewhere for meaning to exist, something acknowledged as fundamentally true. The Atheist has to do so arbitrarily, knowingly arbitrary.
The religious usually anchor their beliefs on a Divinity, usually a Ground of Being, that needs to be taken on faith. This gives bedrock though. It is how value can be ascribed. This is seen as fundamental Reality, as opposed to merely a human 'line' or assertion.
So the religious can use Scientific Method to answer any and all questions an Atheist can. But he can place more trust philosophically, perhaps, in them, than an Atheist would be able to. Further, he can investigate the teleological questions that lie beyond Scientific Method. So it is not that religion cannot answer certain questions - it certainly can, utilising Scientific Method if need be, for they are not about the same things. It is the non-religious that has a more limited armamentarium to investigate and interact with his world.