How much responsibility should an U11 captain actually have?

I’d give the captain jobs that help other players rather than jobs that make them look like a mini-adult coach. Coin toss, helping the team get together before kick-off, encouraging teammates and setting a good example are all useful. They can also speak respectfully to the referee when needed, but I would not make an U11 responsible for managing officials or correcting teammates. Leading a warm-up can be good if the player wants the challenge and knows what they are doing. It does not need to become a performance every week. I’d rotate the captaincy as well. At this age, leadership is learning to help the group. It is not a badge that gives one child extra authority over everyone else.