Well, I've only played RBY, and that was like twelve years ago, so I'm not going to get into the details of it. You just brought up the point that the 'main doesn't even care' for the other games, and I just don't remember Red ever actually doing any 'caring' in RBY either. The rivalry may be more apparent, I don't know, but Red doesn't contribute much on his end.
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"When I was a young man, I had liberty, but I did not see it. I had time, but I did not know it."