The reason to lock them in the coop at night is predator protection. If you feel that run section is safe from predators there is no reason to lock them up.
That looks like it could be tight for six chickens, time will tell. Most of the prefab coop manufacturers use industrial spacing for chickens and that usually does not translate well for us. They are dealing with thousands of chickens and have things pretty automated. Those things make a difference compared to a small backyard flock.
I don't know where you are located or what kind of winter weather you will have. If they get stuck in that enclosed section only you could have behavioral problems. You might anyway but that's not guaranteed. They may be fine behaviorally. Chickens poop a lot and it can build up. The smaller space thy have the faster it builds up. You may have to work fairly hard at poop management to keep the smell down. But again, maybe not that hard. You don't have enough extra room to handle problems if they show up. I value that flexibility, it can greatly reduce stress if something happens.
A lot of people keep chickens in those small prefab coops with numbers like you have. Often it works out though sometimes there are problems.
I think your biggest test will be through their adolescence. Cockerels mature at different rates but sometime between 3 and 5 months the hormones will hit that cockerel. The pullets will still be totally immature and that cockerel will be under total control of his hormones. It's often a very stressful thing to watch, especially if you are not used to chickens. It's quite possible at that time you will decide you don't really need that cockerel. If you can make it through that phase they normally settle down to a well-behaved flock but often watching them go through puberty and become mature adults is not for the faint of heart. That tight space will probably make that process rougher.
I'm not going to tell you that the coop you have wont work, it could. But more space normally makes the chicken experience more pleasant. Good luck!