There are no right or wrong answers to a lot of this. We each have our own unique situations and experiences. There are a whole lot of different things that work.
I find that the chicks are not going to fall out of the nest if you have a reasonable lip on the nest. With a fairly low lip, it is a possibility. Also the size of the nest makes a difference. They are less likely to fall out of a roomy nest compared to one that the hen totally fills.
By the time the hen brings them off the nest, they are usually pretty active. You’d be surprised at what they are capable of as far as getting around. They are living animals so no one can give you any guarantees, but the hen usually does a real good job of taking care of her chicks. I don’t know how big your coop is or how it’s laid out. It’s usually easier for her if she has some room to work with.
My set-up is different to yours. My 8’ x 12’ coop is on the ground. When a broody brings her chicks off the nest, she normally keeps them in the coop for two to four days before she takes them outside. I think part of that is that my pop door is up a bit. I’ve got pavers stacked that they can use as steps but not all newly hatched chicks can maneuver them real well. Most can but sometimes you get one that can’t handle it real well. Once she takes them outside, they spend practically all day out there. With your set-up, the hen may take them out immediately. I don’t know. But with them free ranging, she will have plenty of room to work with. Odds are they will be OK with the flock from Day 1. That’s the way I do it.
There is nothing wrong with separating the hen and chicks for a while. Lots of people do it that way. I really like the hen to take care of integration as you mentioned. Most of my broodies wean their chicks anywhere from 4 to 9 weeks of age. I had one that weaned hers at 3 weeks. They are not really consistent in when they wean them.
From what I think your set-up looks like, I think you’d be fine either way. Good luck!