He wants change the fate of the world
Yea Obito did stated that during that 2 month flash back I remember, but that's not the reason why he doing this.
Obito had crush on a girl he liked who didn't like him back stated several occasions, basically getting friend zone and Rin loved Kakashi cause he was cool. Obito later goes insane when he witness Kakashi killed Rin and breaking the promise of not letting Rin die(which would think Obito would attack Kakashi since he the one that killed her) really nothing that serious cause Rin isn't Obito actual gf. Obito wants to change fate cause he witness his crush die, not because his family has been killed or anything personal, again not a strong reason for Obito antagonist at least to me anyway.
Obito should had never been Tobi in my opinion, Obito was better off like he was previously in Kakashi gaiden.