You may be right, but how do you define all cords?
The ninja story doesn't work here, since for any p1 I think there's exactly one p2 possible. The photographer story DOES work, since he can start at any point, and the mermaid can then be at any second point. I'm not sure about the dragon story, but I'm guessing it doesn't work either.
Of course a circle has no corners to complicate things like that, but it can still have biases. Just because we have found a way to uniquely identify cords does not mean it will lead to a uniform distribution.
This is the sentence that helped me understand the paradox.
Perhaps this analogy will help. Think about how technically picking a value at random from a normal distribution can result in any value at all, extending to infinity in both directions, but that doesn't mean all values are equally likely. Every real number can be uniquely expressed as a number of standard deviations from mean in a normal distribution, but that still has a bias.
So you are in favour of method 2.
But method 3 says: Take a random point inside the circle, and use this as a center point of the cord.
What do you find wrong with this method?
This method has different probability density from the random-center and random-endpoints methods. That's the whole point of the exercise: you get different probability density across the chord space depending on how you select from it.