But I don't think it is cut and dry to say BYU isn't the Church.
For a lot of people, the BYU vs. Utah rivalry is a proxy war between them and the Church. Not for all people. But for some. And it also seems like for many people, BYU and Church are the same thing when it suits their arguments (for example--social issues).
I am not saying this about you personally. But I really don't think you can just separate BYU and its sponsoring organization so easily.