Econdave,
If you can drill and tap the receiver and mount bases on it for less than the cost of a B-square saddle mount, I'd go for it. But I think you'll be looking at somewhere around $50 or $60 to have a gunsmith tap and install bases or a Weaver rail for you. When properly installed, the saddle mount is plenty strong and stiff enough (although it is not exactly pretty or lightweight), and I think you can buy them for about $30. Totally your call on budget and aesthetics of your gun; either solution will give you a rigid, solid mount for your scope--which I think is what you may have been lacking.
And after I posted, I reread your original post and saw your comment about not getting decent groups with iron sights either. As you suggest, that could indeed be attributable to plastic fouling in your barrel. But it could also be attributable to a loose barrel. That's why I suggested double-checking the tightness of your magazine cap against the barrel lug. When the barrel is properly seated, you should be able to grasp the barrel forward of the forearm with one hand and by the receiver with your other hand and not be able to generate any play between the two whatsoever. If you can feel any movement when you twist and tug, you can bet that the recoil is reseating that barrel every time you take a shot--and consequently blowing your groups all over the paper.
Good luck. And remember that it's entirely possible that you may have just gotten a lemon of a barrel. If you follow all of the suggestions offered by others and me on this thread and still can't get decent groups (at a minimum, you should be able to get 3 inches or less at 50 yards), I think you should contact the manufacturer of the barrel about repair or replacement. Is it a Remington slug barrel? Or a Mossberg made for the 870? I own two Mossy barrels, one of which is an iron-sighted 12-gauge rifled barrel for my Wingmaster, and both are tack-drivers (i.e., one-hole groups at 50 yards with Winchester BRI sabots).
You should expect and accept no less. Again, good luck.