After clicking every single day (I was doing this 2 to 3 times a day) (from Sept 2nd when I originally posted) to select the Egyptair, I was finally able to book it. There are again several phantom availability showing, however, one of them was legitimate and made the booking for me. Though it is not convinient, at least it booked it. I will still keep trying to pick my preferred segment and report when I get it booked.