Yes, I saw that site too that said about applying in person in New York. There are a lot of sites that have information about the embassy/consulate but they don't look like the official site. I think the link you sent is the official one. We will contact the embassy to find out the next steps. But I noticed if you just want to message them you have to start a sort of inquiry and provide all of your personal information. I didn't see anything that just outlined the procedure. We will check with them and I'll update you all. Thanks!
I don't think it's up to the airline either. It most likely would be a direct flight though.