For the same reason a cold isn't "just my body." Just like a virus can put your body in a bad state, so too can various stimuli to your mind. You don't just chalk it up to "how things are", you attempt to resolve it. For a cold, your body does that on it's own and you mostly just wait it out. Mental health issues can be similar (I'm just feeling off today), or they might require some effort on the part of the sick person to resolve. Either way, you don't treat it as a personality quirk, as that undermines the severity of the issue (it really is a health issue) and implicitly shifts the blame to the person who is sick[0].
> When you start labelling it mental health, all you're doing is self-diagnosing yourself...
I don't see it as any different than self-diagnosing your cold as a physical ailment. Sometimes people are not capable to doing meaningful work due to their mental state, and people should feel comfortable saying that they need a sick day for their mental health.
[0]: Which, it could be their fault. But ideally we want to describe issues without implying blame. But just like you could unable to work due to a hangover, you could knowingly put yourself into a scenario that you know (or should know) will put you in a bad mental state.
Edit: Forgot the dirty laundry quote. I want to comment on that too.
> It's not controversial, there's even a term for it - airing dirty laundry.
Airing dirty laundry is providing "too much information", saying "I need a sick day for mental health" doesn't strike me as an inappropriate amount of detail anymore than "I need a sick day for physical health". I suppose you could shorten it to "I need a sick day" if knowing any more information is too much.