Why does hydrogen burn spontaneously in fuel cell at room temperature?
You guessed it right that hydrogen and oxygen mixtures will remain stable. Someone in Harvard waited for >30 years and found very little water if any. As you said, there is a energy barrier. What type of materials can lower this barrier? Catalysts.

The electrode materials require some catalysts such as platinum or nickel. Of course, making cheaper and sustainable catalysts is an extensive area of research. If a person discovers a better catalyst which is far more common than Pt, then that person certainly deserves a big award. Copper does not act as a catalyst here.