Interesting!
For whatever its worth, I could not find your John Picton in the Illinois Death Index. It certainly isn't 100% accurate, but it does make me wonder if the possibility exists he was not in Illinois at the time of his death. Perhaps he is listed under a different spelling and I just couldn't find him in the Illinois index.
Are you aware that there is also a Pvt. John Picton listed on the roster in Company D of the 7th Missouri Infantry? This John Picton lists his home town as Belleville, St. Clair County, Illinois.
Now I have no idea if this John Picton is the same Cpl. John Picton who served in the 80th Illinois Infantry, but it sure is one heck of a coincidence (same name, same county of residence...). I don't have any specifics on the John Picton who served with the Missouri unit, perhaps they are two different individuals, but if they are one and the same person, it might indicate a Missouri connection for the John Picton you seek and perhaps he died in Missouri.
Just a theory....
Paul