It'seems to me that the first solution you give to the paradox is correct.

The second is logically flawed. Firstly, it is not the case that "A chord is fully determined by its midpoint." Infinitely many chords have the same mid-point. You probably meant that its length is fully determined, which is true. The second flaw is that if the mid point is near to the border of the circle then fewer chords can be drawn through it than if it is near the centre. This is obvious if you look at the two extreme cases. If the midpoint is at the centre of the circle then the chords sweep right around the circumference. If it is on the circle itself then only one chord of zero length tangential to the circle can be drawn. I think this disposes of the second solution.

The third solution has the same flaw - the chords are not evenly distributed along the radius, as shown above.

One thing still puzzles me, namely that according to my arguments the third method should under-estimate the probability, whereas it over-estimates it.

Can you explain this?

A given point serves as the midpoint of a unique chord.