Organizations that excel at customer service all have something in common, customer service is absolutely pervasive. Everyone understands they have a customer and are interested, empowered, and enabled to serve their customers to the best of their abilities. On the other hand, companies that have a poor customer service culture may have large numbers of folks who really do care and really do a great job, but a few key folks who are slacking can bring the whole organization down dramatically.
As a recent example: Suppose an oil change shop has stellar oil change technicians, BUT, the folks who fix the computers are "not so good". How long does it take until the oil change guys are not stellar any more and are instead spending their time complaining about the computer guys? Just as bad, how long until the the oil change folks realize that no matter how well they do their job, the customer is going to have a bad experience because the computers don't work right?
The precedent for this phenomenon can be found in broken windows theory. In short, this theory postulates that one broken window in a neighborhood can lead to a rapid downward spiral of neglect if not repaired quickly.
Allow any sort of poor customer service will change the social norms in your organization and spread rapidly just like a neighborhood with a few broken windows will quickly fall into disrepair. You cannot have good customer service if your INTERNAL customers are allowed to be treated poorly. Put another way, if employees are allowed to treat each other poorly, how can one expect them to treat customers well?