Thanks. I honestly don't recall ever having seen any suggestion that the AEs derived the eight-ninths circle area formula in the way I suggest - in fact, I have books stating that it is not known how they did it. In the tradition of Dr. T, could you please direct me to the pertinent publications!
The fact that 256/81 is mentioned with respect to the circle area problem and the fact that what I pointed out in the OP actually happens in combination suggests to me that this is a good indication that squaring the circle in this way is how the AEs discovered the eight-ninths formula, potentially leading to pi.r squared.
The point is, I can't prove it - and, as observed, there will always be somebody who declares the suggestion to be emphatically wrong!!