Just sayin', 8sqft X 50birds = 400sqft is one big mamajama coop! But it would be lovely for the chickens. Remember, 4sqft/bird is only the very minimum, and that works out to 200sqft. More is aways better. You never know when you might be forced to keep your birds inside for a day or two...or three...or....In one of my threads recently, someone said 8 I believe as well. Will have to track it down.
So what if you don't have an actual run but the coop is inside a fenced area? Is a run recommend for other reasons?
It sounds like you're saying you'd like to free range your chickens within the confines if your yard which is great if you can. If its 4ft tall they will probably fly over and visit neighbors. Neighbors of all sorts will probably jump/fly over for a visit and a meal. Having a covered run allows you to let your birds outside without you being present to watch for predators. Handy when you get the flu or need to be out of town overnight.
You mentioned that predators don't come near the house. Just because you don't see them doesn't mean they aren't there. They are visiting at night. And with chickens I guarantee they will visit. Plus "friendly" predators will be around even in daytime. The neighbors dog, even the cat, they all love chicken.