matt, if it is a synchrobalance engine, check that mechanism for wear or any sign of dirt or scoring.

I am inclined to agree that the iron-bore Briggs engines are likely to be very durable if you keep the right amount of clean oil in them. However when you say "a few sets of rings" it sounds as if this one has done an exceptional amount of work. I suggest you inspect the valve seats - unless it has been done a couple of times along the way, it is likely to be due for recutting the seats rather than just lapping them. Check the valve stems for fit in the guides, and reset the tappet clearance after you attend to the valves. Take a look at the camshaft lobes, in case they are worn. Naturally, check your ring gaps if you haven't done it in a long time. I hope you have been looking at the fit of the rings in the grooves each time you replaced them. Check the other main bearing (flywheel end) for wear (just looseness of crankshaft in the bearing).