I don't see that at all. He has two arguments. The first is that a common, and distinct, language helps a culture maintain its cohesion and continuity. It's worth noting that many writers from colonized areas have commented on the irony of writing about their experience using the colonizer's language. It's also worth noting the efforts in many areas to distinguish a group's own language from other nearby languages, even mutually intelligible ones. Why would these issues come up if language didn't matter on a cultural level? To many people, loss of language and loss of culture are related. Obviously this doesn't go for everyone, and some people willingly give up their native or cultural language, but no one should ever be forced to, no more than anyone should be forced to give up their religion, way of thinking, or way of life.
The second argument is basically that languages are data points. Studying languages, especially more isolated languages that tend to be in more danger of extinction, helps us better understand linguistics and human communication. If you only know one programming language, it's much harder to learn a new one than if you've already studied 50 programming languages. This increased linguistic understanding helps us study ancient cultures. It also helps us understand language acquisition, which in turn helps us encourage the language abilities of people with linguistic difficulties. It helps us build natural language processing tools. It even helps us develop patience when dealing with people who talk about things in a significantly different way. Something that may sound blunt or crude or inappropriate can be seen in a different light if you understand that different people communicate differently.
And... if you want to get a little "out there"... If humans ever encounter intelligent extraterrestrial life, we will be much more likely to be able to decipher their communication if we've studied 6000 languages already than if we've only studied 60. Even 60 is looking somewhat ambitious for the future of language. It's not hard to imagine a time when the only remaining languages are from a handful of language families: Indo-European, Afro-Asiatic, Niger-Congo, Sino-Tibetan, Japonic, Koreanic, and Turkic.
And if that does happen, the languages that remain will also likely be significantly simplified as languages tend to be when they become widespread. That may simplify communication, but it will make it harder to study communication and how it works.
The author of this piece (John McWhorter) has written a number of truly excellent books about languages. I would strongly encourage anyone with even a passing interest in linguistics to read any of his books that strike your fancy.