Replace the wood you removed from the bottom of the walls and remove wood from the top of the coop and replace with hardware cloth for ventilation. I would still give thought to removing one bank of nests but leaving the lid to prop open to provide extra ventilation and light.Predator proof - whole "courtyard" (enclosed area down side of house between house, garage and shed) is closed off and sealed / covered over the top with chicken wire.
Manually putting them in - well, it's not really possible now as it's fairly open! I put one in - it'll just go straight out. There's no way to close it now I'd removed the door bar - even if I drop that door down, the way the coop is raised up they'd just squeeze out under the bottom.
You need to be able to securely close the chickens in the coop if you intend for them to use it... OR... since they've been fine without going into the coop at all, do you really need a coop? Is your run secure enough for them to simply not have an enclosed coop? You said it's covered in chicken wire so that is NOT predator proof, but if you can predator proof the run that's an option. I don't recall if you mentioned your climate, but if it's moderate, the chickens would do just fine not being inside a coop at all.