Was my coffee actually supposed to be gritty?
Utterly to be expected - the coffee is made with coffee grounds, sugar and water in a small tapered pot. There is no filter. If you want filtered coffee, order something else. There should be grounds on the bottom of the cup. Unless you like grounds, minimal stirring and swirling will help to keep them down there.

This is the coffee that is sometimes described as Strong as [x], Black as [y], Sweet as [z] where x, y, and z are place holders for one of love, hell and death, permuted as suits the mood.