The customer experience is a holistic quality of a site that encompasses many areas of business -- usability is just one of them. For example, Creative Good engagements focus on five areas: strategy, marketing, technology, usability, and the client organization. In each area, we identify how it affects the customer experience, and how to improve it.
So yes, usability is an important part of our work -- but just a part. Running usability tests only is NOT sufficient for customer experience management. How can tests be effective, for example, without a deep knowledge of the strategy of the site? How can the strategy of the site be understood without forming consensus within the organization? Usability processes, refined through the years for *software* products, aren't fully applicable on websites (though Jakob Nielsen's discount usability techniques are effective). To put it plainly, the Web is not software, and the customer experience is not usability.
By the way, I recently explained this perspective to a group of usability professionals here in New York City. I've never seen an audience so close to riot :)