Yeah if you can’t get jack then cheddar would probably be the closest substitute. Also I don’t think Parmesan is used at all. The white cheese that looks like it is a crumble cheese. Queso fresco or cotija would work.
I went to Athens on a business trip in november. I had gyros from the best rated place around my hotel, and it was trash even though reviews claimed it was one of the best gyros in athens.
Also, yeah it's the exact same as trash turkish food