If the existence of a deity did not necessitate the existence of what you consider sufficient evidence, then wouldn't you possibly be requiring impossible evidence for a merely extraordinary claim?
No. If a god exists (ie: a being with unlimited power) then the kind of evidence I would need would not be at all impossible. The fact that I haven't received anything that even remotely seems like evidence for any god's existence leads me to conclude that if there is a god he doesn't want me to believe in him.
Ok.
So you do believe that if a being with unlimited power existed then there would
necessarily exist* evidence that you would find sufficient.
*Though you might not be aware of the evidence.
Under that premise, including the crucial underlined word, requiring evidence prior to belief is reasonable.
You are also correct that while lack of evidence for the positive claim is not proof of the negative claim, it is evidence in favor of the negative claim in cases where the underlined word is present.