Why is the first "Why isn't this more realistic" and the second "Why isn't this more like a TV show?" Why would Klingons speak English in their own ships, other than for the conceit of a television show?
They should speak English for the conceit of a television show. Having to read the subtitles takes away the capability to watch what's going on on screen. It also heavily cripples the actors' capability to deliver the emotion especially considering they already are limited due to the prosthetic makeup.. To me, all of the Klingon scenes felt the same because it was minutes of just dialogue nobody can understand.
Earlier Trek shows were aware of that, which is why the Klingons always spoke English aside from a few words thrown in (and a few exceptions of, say, rituals, which also never were more than a handful of lines).
Also, my first point isn't exactly a "why isn't it more realistic". I mean, if you have a line like "the forecast is radiated with some chance for debris" or something like that - do you really think that kind of wisecracking has a place on a starship? Again, to bring up older Trek shows, there was wisecracking when there was a time and place but there never was this kind of cheesy stereotypical film humour, and while I know that kind of attitude towards dialogue is very common in more modern TV and film, I simply would have appreciated if the tone of the show was in line with Star Trek and not MCU movies.
Federation isn't a traditional military, they are afterall primarily explorers. This was a vessel who's main purpose was to audit some sensor buoys. I'm sure their environment was a lot more relaxed than, say, the fleet admirals.
I know that, but it's military nonetheless.