You said that Ichigo cheated and that that Hougy was responsible for it. If Ichigo cheated because of that, than Ulq cheated too as he used also the Hougy at a certain point.
And the Hougy doesn't make miracles, he just influences situations so that it becomes the thing you want the most, so long you have the potential for it.
And that thing wasn't Hichigo => this is Hichigo
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He's a psycho, but still a 'rational being' that can talk. What defeated Ulq was a thing solely living on instincts.
And even if you say Hichigo defeated Ulq, Hichigo is a part of Ichigo. Hichigo can't live without him, he needs him, not to mention that Hichigo himself doesn't know exactly what he is. So in a certain way it's still Ichigo.
Ichigo never died as he was screaming for help. He was in a near-death state, but in his mind he was screaming. So he never really "died".