Are there special (chartered) Haj flights from the US to Saudi Arabia?

I doubt it very much. Those charter flights have been around from at least the 70s and were financed by the Saudi government to facilitate Muslims from their impoverished neighbors making the Hajj. I heard a great account from a stewardess for what I think was Saturn air lines back in the mid 70s. It was so vivid and exceptional that I will always remember it. The condensed version: This particular flight was the first Hajj trip for the entire crew, and the pilgrims boarded the plane with their belongings, including gear for camping and cooking as well as chickens and other small livestock in cages. It was pandemonium for a while, with people actually wanting to build cooking fires in the aisles, but everyone settled down when it was explained that the trip would be short. It is the end of the tale that I still visualize. The plane lands and the flight crew stares as the passengers embark and march straight into the desert.