"Anecdotal evidence is worthless."
Nonsense. Anecdotes are data points like any other.
"Show me a study where any claimed 'psychic' was able to demonstrate such things in a controlled environment, and be able to repeat the experiment."
Show me a study that conclusively proves the existence of Dark Matter. The fact that you can't doesn't mean Dark Matter doesn't exist.
"One example like this is meaningless."
One might've said the same about the Wright Brothers' first flight. What if we all said that this one example was meaningless, or a fraud, and that the people proposing heavier-than-air flight were all crazy (the news media of the time suggested all of these)? Would you also say that "one example" of, say, man walking on the moon or circumnavigating the earth is "meaningless"?
"Yes, there is a tiny chance it actually exists just as there is a tiny chance that next time I jump in the air I won't come down. What is more likely, that this person has supernatural abilities or that the people involved are mistaken or missing some vital information?"
What is more likely: a powerful God pulls the sun around the sky in a giant chariot, or there is a mystical, magical, and inexplicable physical force called gravity that no one really understands, which causes the earth to revolve around said sun (which is itself a giant ball of Wind which is constantly transmuting itself into Fire in a way which, again, no one can explain), creating an illusion that it is moving around the earth instead of vice versa? As our understanding of science changes, so too does our understanding of what is possible and what isn't. The fact that our sciences cannot explain something like a woman having visions telling her where a dead body is doesn't mean it isn't happening - it merely means that (if it is indeed happening) we do not understand *how* it is happening. Unless you believe that mankind has uncovered all the knowledge of universe (or close to it), you must admit that there are many things that are *possible* that we cannot yet explain.