Great trip report, thanks! We just returned from a long mid-October weekend visit at the Park Hyatt Istanbul, and we agree with your overall assessments of Turkey/Istanbul. We visited the FS Sultanahmet while touring the historical sites nearby, and it looked lovely--though the rooftop bar wasn't available, unfortunately. :-)
We really appreciate the info on Amanruya, as we plan to go back at some point to do a more thorough exploration of Turkey (with Ephesus being a priority).
I can say that on a return to Istanbul, we are likely to focus on the Park Hyatt and the new St. Regis (a block away, opening in Feb 2015). The FS Sultanahmet would be great for a first visit, but being on the new side of the city makes it far easier for dining at the many wonderful restaurants. We weren't terribly impressed as much as others with the FS Bosphorus--since the only Bosphorus facing rooms are super pricey and not worth it IMO, and it felt like a much larger property to us--or the Kempinski/Ciragan--since the main Kempinski building looked horrible and the Ciragan suites are even more than at the FS! (But the Ciragan Palace is an amazing building, so if you can afford those suites or want to splurge, I highly recommend it!)
We have plans to be at Amangiri with friends in April 2015, so perhaps we might see you there!