As I suspected, in the Mexican dish (biftec milanesa) the meat is breaded with egg and approaches the Italian (they eat more veal, so veal milanese) which in turn is close to Wiener Schnitzel of Austria, etc. Mexico, in the cities, used to have more continental European cookery than did the States.
I think that the thick flour gravies developed to add substance to the meals of the poor- much like the anadama or pan bread and molasses of some other poor rural areas.
Upper class southern cookery may have been the origin, however, the chicken-fried steak developing from the once popular Swiss Cream Steak. The following is an old American recipe, from the American Heritage Cookbook.

2 pounds round steak
salt, pepper, 1/2 teaspoon marjoram
Flour
butter
2 sliced medium onions
1/2 cup beef broth
1/2 cup sour cream
2 tablespoons grated parmesan cheese

Sprinkle steak both sides with salt, pepper, marjoram and flour. Pound in, using the rim of a heavy plate. Heat butter in a large skillet (with tight-fitting lid), toss in onion slices and cook to straw color. Lift out onions and set aside. Add steak to hot fat and fry on both sides, over a brisk heat, until well-browned. Add beef broth mixed with the sour cream, grated cheese and cooked onions. Cover tightly and cook over a very low heat for about one and one-half hours or until meat is tender when tested with a fork. Serves 6 (or 2 hungry) people.