It is impossible for Science to inquire the mystery, the same that it is impossible to inquire the mystery of fetal development by using Anthropology. Or to prove the theorem of Fermat or the French Revolution using Science. It is the wrong discipline.
Science is a method to find answers to some MATERIAL phenomena using a specific method (the scientific method, which is imagining theories and trying to validate them with material experiments).
Science is based on philosophical axioms that cannot be proven by science. For example, there is an external world, our senses are trustworthy, the world is regular and these regularities are according to mathematics (laws of nature), the laws of nature are the same in the entire universe, etc.
The discipline to find the ultimate mysteries of existence (which are not material, the same way the theorem of Fermat is not material) is philosophy (which is based on logic), and, more specifically, metaphysics. With metaphysics you can discover some things about God (“natural theology”). Then you can inquire using history if Christianity is true or not.
When you say
“Nothing, as long as you allow Science to inquire into what is currently the mystery”, it is obvious that you believe in scientism.
Scientism is the belief: “Science is the only way to obtain the truth”. But this sentence cannot be proven by science. So scientism is self-refuting: it contradicts itself and, hence, it cannot be true.
“Problem is that established belief systems”
The current scientific understanding of the world is an established belief system, as well as history, for example.
” (religion is a fine example 😀 ) usually discourage such inquiry, often with dire penalties.”
You can find people discouraging inquiry in every discipline. See, for example, the reluctance of modern science in acknowledging the difference of IQ between races. You find fanatic people everywhere.
But this is not the official position of the Christian religion. “Always be prepared to give an answer to everyone who asks you to give the reason for the hope that you have. But do this with gentleness and respect,” (1 Peter 3:15). See also paragraphs 159 and 286 of the Catechism of the Catholic Church.