Women were socially important in Egypt as far as the Ptolemaic dynasty, at least. It didn't fully adopt Byzantine culture until the 5th and 6th centuries CE, and this change was largely fostered by the collapse of the Western Roman Empire. So no, there was no "decline" due to their social system, only a largely unrelated cultural/memetic replacement. (It did fall to the Persians and then to the Arabs shortly thereafter, but by that time the ancient Pharaonic customs had been forgotten.)