I've discovered that every company has a different idea about who is responsible for making sure that customers receive consistently satisfactory treatment.
In some companies, especially smaller ones, a call center supervisor may bear the triple duty of measuring agent performance, coaching agents, and making sure the phones get answered in a reasonable amount of time. These three duties can sometimes be mutually exclusive: for example, when a poorly-performing agent needs to be taken off the phones despite a surge in call volume.
Other companies divide up duties. For example, dedicated trainers or coaches who do nothing but work on the skills of individual agents, and operations managers who make tactical decisions to keep calls flowing smoothly and make sure the technology is working properly.
Some large companies have entire departments dedicated to nothing but quality assurance for customer interactions. They may do customer surveys, hire mystery shoppers, and devise entire incentive programs around trying to improve the customer experience. In a surprising example of corporate wisdom, some of these departments even report up through marketing rather than operations.
But no matter the formal structure, delivering a quality customer service experience always comes down to the front-line employee. And front-line employees are usually pretty good at figuring out a company's true priorities. If making a new sale is more important than keeping an existing customer happy, then the new customer will get more time and attention, even if it costs the loyalty of an existing customer in the process. Similarly, if keeping existing customers is the highest priority above all else, then strange things can happen when this priority gets translated into incentives at the front line.
There is no standard way to manage the customer experience in the same way most organizations consistently divide responsibility for things like finance, sales, IT, and marketing.
In the absence of proven management methods for ensuring customers have a satisfactory experience, a lot comes down to the priorities set by senior management. If they care about customer service, then customer service will be delivered. But if the executives have other things on their minds, then odds are the level of service will be indifferent at best.