Because anti-LGBT isn't the defining issue of conservatism. I couldn't find a conservative/liberal poll on this so we will have to make due with Pew's survey which shows that, currently, 44% of Rs and 75% of Ds are for gay marriage.
This wildly pro-gay rights moderate eats Chick Fil A every other day. I am very aware of their giving and once I tried to edit the Wikipedia page on it to make it more accurate (and was immediately rolled back by an activist editor)