Atheists... why do you believe that God doesn't exist?

[Coffee Jesus]

What you say it true for your example. My point is that saying something is wrong, false, or otherwise mistaken shifts the burden of proof. An atheists go beyond simple disbelief once they assert that their denial is the correct position.

But yes. Burden of proof arguments are trivial like you say. At the same time I think its disingenuous for some to rail against believers and hiding behind a facade of indifference. If they were simply nonbelievers then the appropriate position is agnosticism.

It really isn't as black and white as: atheist, theist, or agnostic.

The scientific method requires that a claim be falsifiable. On its face, this seems to require that some possible observation could flat-out contradict the hypothesis, but this is not so. Scientists use probabilistic reasoning all the time. It's called statistical hypothesis testing. In probabilistic frameworks, people can accept or reject claims to an extent without being 100% certain.

But there is even more to think about. If a machine is programmed to generate a claim at random, what is the probability that the random claim will be true? It's agreed that some metric is needed here, whether that metric be the claim's extraordinariness, how much the claim assumes, etc.
A claim that is without backing usually doesn't have a 50/50 probability of being true. Claims generally begin with a greater probability of being false, even before we investigate. Does that mean we should say claims are false even before they've been investigated? Of course not. For practical reasons, we reserve judgement until the claim has been investigated, and has been found wanting.