We do both.

A) use the language set by the user in their os/browser B) switcher shows the language name in that language

Done, easy, etc. IMO the hard part are great translations and designs that work in languages where every word is a novel. And yet, here we are.

There was a really interesting podcast on the AP style and its entrenched biases - but only available to subscribers:

https://www.canadaland.com/podcast/bonus-who-writes-the-rules-of-news/