Brown the ground beef with the chopped onion in a skillet over medium heat. Drain excess fat.
Add potatoes, carrots, celery, garlic, browned beef and onion to the slow cooker. Stir in broth, ketchup, Worcestershire, mustard, paprika, salt, and pepper.
Cover and cook on Low 6–7 hours or High 3–4 hours, until potatoes are tender.
Melt butter in a saucepan. Whisk in flour and cook 1 minute. Slowly whisk in milk; simmer until slightly thickened.
Pour the milk mixture into the slow cooker. Add cream cheese and half the cheddar. Cover 10–15 minutes, then stir until smooth.
Taste and adjust seasoning. Ladle into bowls and top with remaining cheddar, pickle relish, and green onions.