Rewinding it in my mind, facing and chasing started with the faces covered with powder coat. I faced and chased the top first, in the recommended fashion, then the bottom, cutting until there was fresh metal top and bottom. That tells me the faces were parallel at that point. My tool chases the inside of the head tube at the same time as it faces, so the cups didn't bottom out on something inside.
The head tube is seamed, typical for lower end 531 production frames of that period, so it was most probably a weaker material to start with, but I've faced and chased a lot of old Raleighs and other frames of this quality with no issues.
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If someone tells you that you have enough bicycles and you don't need any more, stop talking to them. You don't need that kind of negativity in your life.