I came home and tried to see if the gun would fire in several different open positions. It would not. It would only fire after closing the bolt completely. You could hear and feel a definite click when the bolt closed. It would not fire until you heard and felt the click. I talked to my brother today. He had this happen in a 16 gauge auto last summer. The first time it happened, it was when my 10 year old nephew shot it. My brother put in another round (probably not the smartest thing) and it did the same thing. He was shooting the cheap Winchesters as well.
It's your gun, and your call, so if you are convinced it's the shells I wish you the best. I just wanted to make sure you had all the information you need to make your decision and not hurt yourself or others, which I feel I have done. Dropping the trigger group on a Rem 870 is literally a 30 second job and would tell you a lot, and I am sure there are even videos on youtube showing how to do it, but I understand you not wanting to mess with it since you believe it is the shells causing the problem.
I personally don't understand what would make a "cheap" shell cause the bolt to open, but that is just me. Is it the lack of powder? Cheap wad? Aluminum base? Light shot charge? Why wouldn't a good stout loaded shell, operating at a lot higher pressure, cause the same thing to happen?
Good luck and be safe my friend.