Pat all meats dry with paper towels. Heat 2 tablespoons olive oil in Dutch oven over medium-high heat. Working in batches, sear pork ribs for 3-4 minutes per side until golden brown, then transfer to a plate. Repeat with beef short ribs and Italian sausages, browning all sides.
Reduce heat to medium and add remaining olive oil if needed. Cook crushed garlic for 1-2 minutes until fragrant, then discard. Add chopped onions and cook 5-7 minutes until softened. Add tomato paste to center of pot and cook 2-3 minutes, stirring constantly until darkened.
Pour in red wine and scrape up all browned bits from bottom of pot. Let simmer 2-3 minutes until reduced by half. Add crushed tomatoes, tomato puree, beef broth, basil, oregano, salt, pepper, sugar, and red pepper flakes. Bring to a boil, then reduce heat to low.
Nestle all browned meats back into the sauce, making sure they're mostly submerged. Partially cover pot with lid slightly ajar and simmer gently for 2 hours, stirring every 20-30 minutes. Skim excess fat from surface as needed.
In a large bowl, combine ground beef, ground pork, breadcrumbs, Parmesan, eggs, minced garlic, parsley, salt, and pepper. Mix with hands until just combined. Form into 2-inch meatballs. Brown meatballs in a skillet with olive oil on all sides, about 6-8 minutes total.
After sauce has simmered for 2 hours, gently add browned meatballs to the pot and spoon sauce over them. Continue simmering for another 1.5 to 2 hours until meats are fall-apart tender and sauce has thickened. Taste and adjust seasoning.
Remove pot from heat and let rest 10-15 minutes. Skim excess fat from surface if desired. Serve gravy over cooked pasta with meats alongside or on a separate platter. Garnish with fresh basil and grated Parmesan cheese.