So if I understand correctly the idea is a small protected area (within an existing run) with an automatic door to let them access feed during the day? And the goal is to allow you to avoid having the kids feed them every day?
Hmm. How many chickens? My first thought is that in order to create a big enough area where they can all eat this food area will end up being pretty large. Like, in the morning my chickens all race out the door to get to the grass... But your chickens are trying to get food, so if the enclosure you're making is too small, they will simply stop when they get to the food and create a huge traffic jam at the door. Nobody can get in or out, and only a few chickens are eating.
My honest gut reaction would be that the amount of effort to make this enclosure may be about the same to simply secure the run. But, not knowing size, it's hard to tell.
So, trying to solve for all of this, I'd probably start by deciding the container. Maybe a diy trashcan autofeeder with holes cut out at the bottom (you tube has examples). This would allow you to easily remove lid and refill. But then you need to secure it, which is where it gets difficult. You effectively need to build a decent size 'run' around it. Cattle panel circle (16ft so 5 ft diameter?). Hardware clothed, skirted, and topped with chicken wire? Can near edge furthest from entrance so chickens don't jam? Chicken wire top goes around can so can easily feed.
which honestly seems like enough work you might be better off just securing the run. I think that's the basic issue here - you need a rather large, very secure, feed enclosure that is big enough to handle your chicken quantity comfortably accessing food when it opens . In other words, a run.
the other option would be multiple self secure feed containers that open on a timer, but after a quick look I didn't see any real options online. But some type of autofeeder might be the way? Ive seen some ideas online but nothing id consider secure in and of itself
Or perhaps I'm misunderstanding the problem