I rebuilt a Chevy 195 upright-six back in 1976. Had 100,000 miles on it. I didn't touch the main bearings, what with it being a 7-main-bearing engine. I put in rod bearings, .020 over rings without any work on the bores, new lifters, ground and lapped the valves and put in valve oil seals. That was a big deal because it didn't have them from the factory and it was all oil-carboned up from leaking down the valve guides. Didn't mess with the valve guides otherwise. I drove it for another 150,000 miles before it sucked the head gasket and I gave it last rites.
But I think you can blow-by the rings or the head gasket and get the same results. You're pressurizing the oil system with exhaust and it will come out the breather and into the intake. Since the head's been off recently, and it's easy to fix, I would start with the head. New head gasket, and I would glue that sucker down with a good head gasket sealant. Those sixes are notorious for head leaks, both the Chevy upright sixes and the Mopar slant sixes. When I rebuilt mine, believe it or not, I sealed the head gasket on both sides with regular Perma-Gasket gasket sealer, which is not the approved method (no kidding, right?), but, like I say, I got 150,000 more miles out of it before it sucked the head gasket.
Head gasket and sealer is cheap and you can replace one in a couple hours. Rings is another whole issue. I would change out the head gasket and seal that thing up, both sides of the gasket, as the first shot at this problem and see what you get. And while you're at it, put umbrella valve oil seals on it.