k-cup and Non Political,
You are confusing evidence and burden of proof. The absence of evidence not being evidence of absence (without other evidence demonstrating why evidence ought to be expected in this circumstance) is self-evidently true logical axiom. It doesn’t matter if there is or is not other evidence. That doesn’t mean claims for which there is no evidence are true. It simply means that there is no evidence in which to make a determination either way (in real life there are very few claims for which there is no evidence for or against). But the burden of proof will be on the person making the claim. So you can claim that everything is made out of invisible unicorns but the burden of proof (to the extent anyone cared to ask) would be on you. To sustain your burden, you would have to present some evidence.
No faith or belief in the world has lasting power without evidence. We may also have evidence against their claims, the evidence may be fabricated, or we may dismiss their claims because prior assumptions, but the claims themselves will have evidence to support them.