I couldn't find anything on the net about whether Mossberg ever made a 500 that is 2 3/4 only.
However, I doubt that they did.
What your problem may be, is that you are not using enough force to open the action. A new Mossberg is stiff, and you might have to cycle the action a few times before things work properly. Shells exit the receiver rear-end first, and rotate. So, the slot doesn't have to be the same size or bigger than a spent shell.
While your shotgun may have Extractor and stiffness issues, it sounds about right to me.
You could also try to remove the paint on the left side of the barrel. Mossberg prints what shells can be used in their shotguns on the left side of the barrel.
Since the 500 was made since 1961, it is hard for me to believe it doesn't take 3 inch shells. Mossberg would gain nothing by producing a 2 3/4 500. Even back then.
I did find out that the 500A was any 12 Gauge, the 500B is 16 Gauge, the 500C is 20 Gauge, and the 500D is .410.
Don't baby it. Rack the shells hard a few times, it'll loosen up.