So, do you think all facts require some foundation of belief?
Yes, I think so. When (working from an idea of Descartes) I realized that I operate on the three assumptions I mentioned, and that I'd never noticed them because everyone else acts as if they share those assumptions, I found it very clarifying. They share the quality that it's not possible to demonstrate that they're correct without first assuming they're correct.
But those assumptions (beliefs) are in place because they work, and I suspect they work because evolution is ruthlessly efficient about how only the survivors breed; so they don't need to be known consciously, though if they are, you see they're assumptions of necessity.
Whereas assumptions about ─ beliefs in favor of ─ the supernatural are neither supported by evidence after they're assumed, nor necessary for survival and breeding.
As perhaps a qualification to that last statement, I'm inclined to think that religion and supernatural beings arose usefully from a combination of our evolved traits ─ first as gregarious primates who benefit enormously from cooperation, originally within our tribal groups, and therefore tend to survive better if our practices reinforce tribal identity and unity (along with kin, language, customs, stories, heroes &c); second as curious creatures whose instinctive response to the unknown is to attribute a reason (eg with lightning, dreams, luck at hunting, death, drought, plague &c); and so on.