People already paid for the old-gen-console and don't buy another one (except the first breaks). So no, I can't follow your argumentation.
But anyways, fine, we have different definitions of gen. I accept that. Your definition says the XBOX won gen 6. Fine, keep your opinion. I don't see any reason to fight it, but for me PS2 won gen 6 not gen 5.
But that doesn't matter, because in the end the Wii outsold the 360 and the PS3, which were its direct competitors. So even if we disagree that the Wii was in a separate category from its contemporaries, it still outshined them.
But for the first part, try to follow. In one case, you have to pay the price of a whole new console (2 to 3 hundred dollars), in the other (the competitor alternative), you would only have to pay for the add-on (30 to 60 dollars). It's not a complex argument I'm making.