Your mountain analogy actually exposes the flaw in your argument. “No successful climb has been demonstrated” does not prove nobody climbed it, but if the claim is “someone climbed it,” lack of evidence gives us no reason to accept it. Likewise, if specific religions make specific supernatural claims, “you haven't disproved every religion” does not give those claims positive evidential support.
You also keep switching between “religion” as every cultural practice and metaphor and religion as factual supernatural claims. I don't need to prove that prayer, ritual, art, or metaphor is false because those aren't necessarily truth claims. The proposition I can reject is the factual claim that a particular god exists, revealed a particular scripture, performed particular miracles, or guarantees particular supernatural outcomes. And your “one true religion, therefore not zero” argument is a false dichotomy: the possibilities include one, several, or none. Religious diversity is enormous, with Pew's 2026 analysis identifying seven major categories across 201 countries and territories, while even within traditions there are major disagreements about doctrine.
Most importantly, you're demanding disproof when the rational standard for belief is evidence proportional to the claim. An unfalsifiable claim isn't automatically false, but it also isn't entitled to belief merely because it cannot be falsified. Philosophical accounts of falsifiability specifically distinguish claims that can conflict with possible observations from claims insulated from such testing.