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What does "Set Sort Tag" do?

It is persistent ... I figured that's what you were going for. It will maintain the sort order of that playlist on whatever tag you choose. If you make changes, it will reorder your tunes accordingly. The only difference SHOULD be that Set Playlist Order is a one-time thing, but it sounds like there may be an issue.

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any items that sort equal on the current key remained sorted

I never wrote that intentionally, so that sounds like a side-effect of whatever the sort algorithm was doing. It should have been the case that if you sorted on Year, then clicked on Artist, it should have resorted everything. It was probably a coincendence of implementation that previously equal fields would not reshuffle. I think QuickSort by default does not guarantee that two equal key comparison won't move.