I think you are right that the chick flew up to the top of that fence just to perch, then hopped down on the wrong side. That’s pretty common when you have a solid rail on top. They can fly quite well if they are properly motivated. For chicks the age of yours, 6’ is not much of a challenge. My full sized adults easily fly up to my 5’ high roosts. I’m sure they could go much higher if they really wanted to.
I see three options for you. One is wing clipping, but since I don’t do that I can’t speak from experience.
You can put a top on it. At 16 feet wide that will present challenges as you mentioned. Whether it is some type of solid roof or just bird or deer netting, you want to be able to walk in there without hitting the top even if you don’t go in there much. Netting will sag. You need it strong enough for snow/ice load and maybe even wind if it is solid and maybe snow/ice for netting. You can put in longer posts and put a ridgepole down the center of that maybe 2’ higher than the 6’ sides and use that to raise the netting or make the solid roof drain. A flat solid roof will provide shade but will leak. You need enough slope for the water to run off. And it needs to be pretty strong with even an 8 span. Don’t even think of spanning the entire 16’ with a solid roof. Lumber heavy enough to span that would be too heavy to handle, especially at height, and very expensive.
Something I’ve done is extend the wire up the sides so it is above the solid rail. If they don’t see something solid to land on, they won’t fly up. Cattle panels should be 52” tall. You buried them a few inches so let’s say they are just under 4’ tall. Get some 2” x 4” welded wire fencing 5’ tall. Other stiff 5’ tall fencing will work. Attach that to the top of your cattle panel maybe with hog rings, then roll it up and attach it to your top rail at 6’. It will stand up under its own stiffness and give you just under an additional 3’ of fencing height. There will be no top rail so they should not try to fly up there.
An added bonus, if a predator like a raccoon tries to climb it, the fence should bend back under the weight of the animal making it real hard for it to get over the top and into the run. They will still be able to get in over your coop and maybe the gate so it is not perfect, but it’s a help.
The gate will be a bit of a challenge. I’d suggest attaching 2x4’s to the posts to extend them up higher and fencing in above the gate. I’d attach a horizontal at the bottom of the gate posts just above the gate but not at the top. That gives you something to attach the bottom of the wire to so it doesn’t get in the way of your opening the gate, but I see a potential problem down the road.
I don’t concrete posts in either but when I tamp them in, I use rocks firmly tamped in with a 5’ iron bar made for that purpose. If you firmly tamp in rocks at the bottom, fill the middle with dirt also tamped in, then firmly tamp in rocks at the top, that post is not going anywhere, just like concrete. There is a bit of a learning curve but I have no shortage of rocks and Dad made sure I went through that learning curve growing up. But you said you used dirt, not rocks. With the weight of that gate, I expect that post to lean over time. If you can attach a horizontal just above the gate to the posts on those extensions you can brace it to help stop that leaning. It looks like you could run that horizontal on over and attach it to the coop also for added strength. Taking out a bit of dirt and tamping in rocks at the top can help even now. Just get the rocks tight.
I think you did a good job on that. I like the way you used cinder blocks under the gate to make a sill. That will help keep predators out and chickens in. Gates are often the weak spot in the run.
Good luck with it.