My family was like this too, really fixated on the death of one and the birth of another. I honestly can't claim to understand it, and it really upset me for a long time when people treated me like I was killing the old person. Because for me, it wasn't a death or a birth, I just wanted to stop the s* in my head, and transition successfully did that for me.
Hormones really facilitated that process for my whole family. My grandma, who was the most opposed to the transition and the most adamant that she would never use my chosen name, even told me "it's so much easier now that you don't look like a girl, because before I felt like I had to explain to my friends that you were a man in a woman's body, and that made me uncomfortable. Now you look like a man and I don't have to get into it, if they knew you before I just tell them you're not that name anymore." Same thing with my parents, they've both told me that I am just SO much more stable and better to be around...it sort of assuaged their concerns that transition would screw me up rather than make things better for me.
I think part of it is them knowing their "new" child will come out better for it, you know? They worry the transition will make the child they knew a different person or affect their physical and/or mental health negatively. But I feel you, it sucks when you know you're being yourself and others treat you like you you're not the same anymore.
