That theroy has been posted before.
The one thing about the Burkes that I wonder is whether the children are of both or if they are like stepchildren.
Another thing is that Tybalt mentionned he comes from a poor background: it could be that he is not Stephen's son and only got rich because of the marriage of Jade, his mom. That would also explain his attitude towards wealth.
Regarding Sage, one hint going in the direction of her part of the family is her interaction with Isabella who seems surprised at her intellect. There are many reasons for that but I think it could partly be because of Jade. Another thing is how Chad is treated: Burke gave him a pass for the assault and that could be because of Sage.
Finally there is the whole prostitution ring going on: as far as we know Sage is in the dark. Maybe Quinn thought she would disaprove on principle but it could also be because Stephen is a client.