Is there such a thing as cheep food in Branson. I am not talking about fast food.

Is there anywhere a family of 4 can sit down and eat a good meal for $20?

Sure. McDonald's. But if you want good food you won't find it for 4 people for $20.

Danna's BBQ would come pretty close to that price and the food is excellent. Have water instead of soft drinks,that is what adds to the cost to a meal.
At Cracker Barrel they have an egg sandwich on the breakfast menu that I have had many times. Have them hold the mayo, grill the sour dough wheat or white bread and have the eggs prepared your favorite way,over hard is my choice. You can have the fried apples or hash brown casserole, or ask for plain hash browns. Ask for some apple butter for the toast. It's 2.99 at our local Cracker Barrel yesterday.