>Does Information Theory shed any light on Bertrand's
>Paradox?

Indeed, looks like it does.

>Thus it would seem to be that the solution where the
>midpoints are distributed uniformly over the radius and the
>probability becomes 1/2 is the solution with the "most
>random" choice for the cord.

The reason the situation is dubbed a paradox is that there are more than one plausible solution. Yes, Jaynes' and your arguments show that in one sense or another one of the solutions is more plausible or natural than the rest. They do not negate the presense of three solutions.

I do not know, perhaps these arguments should apply to the definition of probability so that Bertrand's chord problem would have a single solution.