Back in the day.....
We had a pigeon "cote" with a high take off and landing door/platform, raised nest box, branch roosts and food shelf and the whole thing up on stilts. I never saw a hen in it and yes they could fit through the door.
I think a big factor not listed by a few of the people posting is exactly what type of chicken they are keeping for those cotes. Game birds that are light and can fly might truly enjoy the opportunity to fly up several feet to get in and some Banties too (those two types did like to roost in the trees)
But other chickens are just too big and too heavy to be forced to fly. The LF chickens most of us keep for eggs in our back yards are very different from their ancient ancestors
Our chickens have the range of the garden, they will hop up on saw horses, the compost bin wall and a stump of a tree. And of course over the fence/gates to the veggie garden. But I do not see them fly up to the open bird feed plate (4 feet off the ground) or up in any of the trees. On the other hand, they have their favorite trees to hang out under, all with low branches they could easily choose to perch on. Our hens, LF, seem to prefer the ground. Even hidden nests are located under bushes. If you saw our lot, you would realize they could pick many places up high to hide their nests away. They love to run down the hill flapping their wings to help them race to the bottom of the yard, but they do not try and fly.
That all being said, our new coop is raised, they have to hop up about 3 feet or use the chicken ladder to get in the coop at night. They love the high roosts (a good 6+ feet off the ground ( 3.5 feet from the floor of the coop) We find a few of them in there at odd times of the day choosing to nap in there rather than a tree branch. Before you say that is cause they were raised in that coop and that is why they pick that spot, the new coop has only been used by them a couple of months.
My point; coop or cote, seems the bigger question is the type of chicken using it.