My roo, who was born this winter, chases cats away no problem. Well house cats. We have bigger cats where I live and if anything he can't fight shows up he sends the call an they run to cover. Have you maybe shown them they can travel through the opening? When I first opened my pop hatch I poked my head in a few times. The ladies would look out the door but not come out for a while. Then one morning my roo felt adventurous and explored the run. Now they all spill out every morning and huddle up at night.
Oh, they have a wide open door. Our coop is really just a wooden shed. We've actually taken them out a few yards away, and given them treats. They hang out as long as we're outside, and then head right back to the coop. The cats have been known to eat tender, young cockerels and pullets, but haven't attacked any chicken older than 8-10 weeks. I guess I'm just going to have to be patient, which is not my long suit.
Update: The chickens have been happily foraging outside the coop for the past few days. Today, for the first time, they ventured into the front yard (about 50 feet away from the coop). I've decided my scaredy chickens are merely cautious and smart chickens. They're very gentle and tolerant of the daily handling I give them. I love my tiny flock!