In this case that would make sense, since there's no way to trace a shooting to a shotgun through ballistics AFAIK.
“It is usually stolen guns which have serial numbers ground off.”
It is possible. It is just as possible that it is one of the millions of shotguns made before 1968 that were not required to have serial numbers.
The requirement for .22 rifles and shotguns to have serial numbers was part of the 1968 gun control act.