Agreed, listening carefully to your partner is key to resolving a conflict. However, by the same token, those customer support lines exist to help customers get their questions answered quickly and without difficulty. The fact that the options or source of anger have changed doesn't mean that the customer should ignore what they're being told - it just means they need to take it in context rather than recite it by heart.
The idea of memorizing the sequence of digits in order to get to the gardening department's delivery service's hours of operations on Tuesday in English is a bit much, yes, but that does not invalidate the point made in the original essay: Customer support lines can be time consuming and frustrating. The fact that the system tries to be as efficient as possible does not excuse its often lack of user-friendliness.
Moreover, the retort implies that understanding your partner's needs is a straightforward process that begins and ends with listening carefully. Unfortunately, this is not always true. It takes more than just listening to understand another person; it takes curiosity, empathy, and patience. It isn't enough to simply hear what your partner is saying - you have to actively engage with it in order to bridge the gap between two different perspectives.
So while it's true that communication is key to resolving conflicts with a partner, it's also true that customer support lines can be unnecessarily complicated. UrRong in thinking they are the only solution - sometimes a hug and a reminder of how much you care can do just as much as memorizing a bunch of options and numbers.