No, chickens do not NEED a run if they have the necessary coop space. But like many people already said, chickens just seem "happy" outdoors, and a well-built run keeps them safe from predators. I'm getting six chickens in July, and I have a run in the works right now that's 8 feet x 16 feet (128 sf) that will provide around 21 sq. feet per chicken (in addition to their coop, which is small). It will have swings, perches, food, dust bathes, and other enrichment sort of things to keep the chickens "happy". Content chickens with plenty of space don't have time to bully each other as much.
Yes, being outdoors in a run does open up possibilities for the chickens to get diseases from wild birds, but does keeping them cooped up do any favors to their immune systems? Exposure to diseases can make chickens more resistant to them.
Plus, they are descended from wild birds, so they've got the love of the outdoors in their blood. It's probably instinctual for them to want to be outside, sort of like a house cat that wants to go outdoors. But just like people build "catios" that are large patios covered with hardware clothe that allows the cats to go outside without running wild or possibly getting hit by a car, so chickens have runs that let them go outside to enjoy the fresh air without being taken by predators.
By the way, I am speaking more from a urban chicken keeper perspective, not as a farmer with lots of land. I live on a small quarter acre lot in the middle of the city. My birds can't free-range, or else every stray dog, cat, raccoon, skunk and hawk in the city would simply help themselves to a free chicken dinner (and perhaps some of the neighbors would too!

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