I think what would have been called "country style dinner" in most places in the U.S. was indeed kind of like Hamburger Helper, but with somewhat more natural (or at least identifiable) ingredients.
One version might be ground meat, noodles, tomato sauce, butter, salt pepper, and shredded cheese. Cooked in a skillet (I think that's the "country style" part), browning the beef, adding all but the cheese, cooking until the noodles are done, and adding the cheese.
Most one pot dishes I see called "county style" are skillet dishes of meat supported by a mass of other ingredients that more or less solidify. But it can also be baked. A country style chicken dinner might have diced potatoes, vegetables, milk, cheese, chicken, and something to help thicken it.
Not very inspiring cuisine, but easy, not too expensive, reasonably good, and filling. I guess Down East, the New England boiled dinner occupies that place.
"Kitchen duty is awarded only to those of manifest excellence..." - The Master, Dogen