On the surface, this idea sounds very appealing. Who wouldn't want to eliminate waste, eliminate error, and make their processes more effective? The goals of this idea are admirable and useful. Where they apply, they are great things.
This idea does not apply everywhere, however, and trying to make it apply everywhere is a very bad idea. Start with one of the most basic concerns: this idea applies to mechanical processes like manufacturing, but it does not apply easily to other processes. Imagine, for example, trying to apply a focus on removing variation to an industry with a lot of variation, like medicine. A doctor can only apply certain tests, because people can only have certain diseases? That logic is laughable. The push-back against standardized testing shows how removing variation creates trouble in education. Children aren't all the same age, and don't mature at the same pace. Even if they are in the same grade, they don't learn the same way. That applies to adults as well, and so any organization focusing on customer service would get in its own way if it insisted on removing variation.
There are also more complex objections to applying Six Sigma methodology to organizations, even those which focus on mechanical production, like factories. One of these is that Six Sigma applies well to mature processes that are well-understood, but doesn't apply to developing processes or development processes, like innovation or research. To spell out this objection, it makes great sense to try to remove variation when you have an assembly line and are actually producing, say, cars. It makes less sense to remove variation when you are trying to create new ideas. Stages like brainstorming require variation, so applying Six Sigma too broadly throughout an organization or too early in a process could cripple an organization. Six Sigma works well with mature processes, not developmental or emerging processes.