Maybe the flaw isn't in Bayesian analysis per se - maybe the flaw is in expecting all claims to be falsifiable. Any theory of reality will ultimately depend on some unfalsifiable claims, which are really just the axioms.

Bayesian analysis treats the earliest priors as axioms, because they are considered to be beyond the scope of analysis. This would be the "incompleteness" described in Godel's Incompleteness Theorem.

Tl;dr: Frequentism, reductionism, and Bayesian probability - are all tools we can use, and you helped us see their inherent limits.

In war, this manifests as a Pascal's Mugging.

Is there a name for the following paradox:

Country A says "We have to fight Country B. Yes some people will die. But if we do nothing, Country B will attack us and 10 times more people will die!"

Country B says "We have to fight Country A. Yes some people will die. But if we do nothing, Country A will attack us and 10 times more people will die!"

Even though both countries seem to want as few deaths as possible, their actions combine (and escalate) to cause more deaths.