If you ever regard wide-audience advertising, most of it is devoted to status games; imx breakfast cereal ads are the only ones that seriously resist such low-effort interpretation.
The issue seems to be more subtle. Nobody is trying to explain all of human behavior with primate studies. Rather, to establish that there are still traces of our animal nature visible in human behavior.
We tend to believe we are no longer influenced by our animal nature. But without accepting that part of our construction, it not possible to understand human behavior, because that part is essential. Usually, statements like "primate studies cannot explain all human behavior" are used to de-legitimize some studies, in order to protect the taboo.
If you think that not all of human behavior is influenced by our animal nature, then certainly you can name some behavior that you believe is influenced by our animal nature. So start by asking yourself, what is such a behavior.
I would like to think that we have evolved new ways to live in complex societies.
We have non-primitive languages for instance.