Actually, you had the right idea, but asked the question wrong. The 5.56/.223 situation is related to the way the chamber is cut in the barrel and the the amount of leade. 5.56 chambers will always handle .223 ammo safely, but the reverse is not always true. But IMO, the ammo question has little to do with the problem of the "trigger not resetting."
I put that part in quotes because of something you said earlier. You said that moving the selector to safe, and then back to fire, would enable you to discharge a round. That indicates to me that the hammer is being moved to the cocked position, but for some reason the trigger group in your rifle is malfunctioning. Working the safety causes the trigger group to return to normal function.
If it were my rifle, I would be looking at the trigger group.
Regarding having to break-in the rifle, none of my ar15's have needed that. I did follow the factory reccomended "break-in" on a Bushmaster, but that was for seasoning the barrel to ensure long-term accuracy. They have all worked fine from the first round. I'm not saying that a break-in period is impossible, just that I have not encountered it. I do have one
lower with what I consider a crappy pull, but it's always been reliable, just not enjoyable.
I hope you are able to resolve your issue.