My family recipe is just like the OP except mushrooms (canned or fresh), sliced onion are browned with the steak first. Instead of onion soup mix and tomato sauce, a can or two of stewed tomatoes are used. If enough flour is pounded in, this makes the most wonderful gravy.
Lately, I've been using chuck steak in lieu of round for both increased flavor (the extra fat and juices help make the gravy beefier tasting) and the more moist texture of the meat.
Dang! Now I have to go to the store in the morning, as I've given myself a swiss steak Jones