Trains aside, the British carrier BMI fly from the US to Aleppo and Damascus via London Heathrow. It will be worth checking out.
Overnighting in Istanbul might also be a pain in the neck (and cost you) once you start figuring out the connections you have to make. You will need to book the sleeper in advance if you want to be sure of a bed and therefore pick up your tickets. If you book online with Turkish railways, you need to do that in Turkish...
Well, BMI is not the only airline linking the U.S. with Syria, with 1 transfer - others include Austrian, Malév ( Hungarian ), Egyptair, Royal Jordanian, Turkish Airlines, Czech Airlines... ( Austrian, BMI and Royal Jordanian ) also serve Aleppo, by the way. Austrian and BMI are the only ones arriving and leaving at a convenient 'day' time - others do so unpleasantly in late evening or at the middle-of-the -night, which is not very amusing if you get into an unknown country, an have still to reach your hotel !
But the train solution is definitely a pleasant and adventurous way of travelling !