It doesn't change my point that without LBJ this stuff would not likely have passed. And if you look at the numbers the vast majority of both parties ended up voting for it.But my point is that it wasn't the Democratic Party that wouldn't let it out of Rules, it was Smith and a small cadre of southern democrats. Johnson and Bobby Kennedy were working their way around Smith, and as you correctly stated, Johnson used his office to force Smith to give it up.
I agree that Nixon's Southern Strategy alienated those of african descent from the R Party. But saying that I was largely wrong because of such a point is a bit ridiculous. There was far less opposition to the legislation from the R Party than there was from the D, and those in positions of power were from the D Party. Without intervention from LBJ the law would have died in committee. It certainly wouldn't have been opposition from the R Party that would have killed it.