"Slippery slope" arguments rarely hold merit. As intelligent people, we have to argue each point on its own merits.
George Washington himself owned slaves. It's a sad fact that practically all of our American institutions owe themselves to the mistreatment of primarily black people. It will generally not be possible to untangle every point. If a point has other positive qualities, then those have to be considered, too.
However, here we're talking about a song. True, it's a song that's represented millions of alumni and Texas fans for over a century. It's an institution unto itself. It's still, though, a song. If that historical institution divides a Longhorn of color and makes him or her feel less than included, then we should change it.
I can't answer the question of how feelings of racism are attached to it. Only those affected by it can do that. Thus, I'm calling for them to understand what they're asking, and to be really sure change is needed.