As I've said, it makes zero sense to assume that the crew speak different languages and have the translator constantly active. So I choose to interpret that scene not as the translator deactivating, but rather as the translator activating and malfunctioning, translating the crew's English into other random languages. It's the only way to make any sense of it.
Yeah, i'm on board with that as well. DSC as a whole didn't make sense, but at least this one i'll chalk up to the universal translator kind of randomly translating what people were saying into all manner of languages.
It is quite clear the primary language of the Federation... or at the very least, Earth and Starfleet, is English. It makes an amount of sense in-universe.
Post-WW3, the western nations seemed to be doing better than the eastern nations, given that the USA is involved with sending out warp colony ships to other planets while a distinctly eastern looking courtroom is from the "Post-Atomic Horror" in the same timeframe, and add to the fact that even in the modern day real world, English is fairly commonly spoken in many countries.
So if English becomes the predominate language spoken by Earth's space agencies, and space becomes super important, it stands to reason that English would become even more prominent.
When we encounter aliens, there's a somewhat odd observation I can make that they seem to be better at learning English than we are of learning their languages. Take TNG "A Matter of Honor", Riker certainly doesn't know how to speak Klingon, but every Klingon on that ship can speak English... and that's implicit in the episode, "Speak their language".
So if Earth primarily speaks English, and aliens find it easier to interact with Earth by just speaking their language... English becomes the standard language.