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My understanding is you want the weight down. You just have to make sure that when you move the weight, it's not just spinning on the shaft and that the shaft is actually turning.
I've got one in pieces off a 1959. I thought I was gently tapping it off the three bolts with a small hammer. I broke it in half. But the flapper is in good shape. Maybe the two pieces could be brazed together? Yours for shipping costs from Oregon if you want to mess with it. The flapper moved freely before I broke the cast iron surround.