There's no alternative: all belief systems must have some axioms that people agree on "just because", as if the axioms required justification, then the system used for justifying them would also need to be justified, ad infinitum. It's a bit like Gödel's second incompleteness theorem: a moral system cannot prove itself to be moral.

What we can do is try to understand and compare the axiomatic systems upon which belief systems rest. Marxists believe in the labour theory of value (very roughly, "value" comes from time spent working). Austrians believe "value" is a form of ordinal utility with which intrapersonal utility comparisons cannot be made. Mainstream economics believes "value" is a form of cardinal utility with which intrapersonal comparisons can be made.

Within the context of one of those systems, if somebody argues for something that doesn't maximise what the system values, we could say it's unscientific. If somebody makes an inconsistent statement, we could argue it's illogical. But if somebody makes a statement that is consistent with maximising value as defined by that system, we cannot argue it's unscientific just because it doesn't maximise value as defined by our own system, because science cannot say whether our own system is morally superior.

I think you're a bit confused. The "not allowed to question this ideology" problem is independent of whether the ideology has a supernatural component or not. You weren't allowed to question Mao when he ruled China or Pol Pot when he ruled Cambodia; in many circles you aren't really allowed to question transgenderism or other ideas as well.

On the other hand, there are long histories of philosophical and theological inquiry in the Christian tradition; from what I understand the same is true of Islam. There are certainly times and places in both camps where small deviations were perceived as heresy, and perceived "heretics" were persecuted or killed; but as a Christian, I am as opposed to that as much as any atheist can be opposed to the persecutions of Mao.