I guess the question is why you need them to know. I can think of a million other legitimate medical reasons that I'd never put in an email to my team. Why is this any different?
Also, you should consider that if everyone gets away with being consistently vague, you set a team standard that says people can stay home whenever they feel bad, for whatever reason, and it becomes harder for the company to legally crack down on someone who's doing it for emotional or psychological reasons.
Feelings aren't medical reasons, they're core to being human. If we cut that off from how we're allowed to be in the workplace, we can't really be ourselves. That causes problems. Imagine if people wouldn't let each other breathe properly. That's analogous to where we are with emotions at work.
Integrating feeling into the workplace doesn't mean sharing every emotion, any more than you'd share every thought or every story about your life. It just means being able to accommodate that information when it's relevant and respond appropriately, instead of seizing up with a taboo reaction. Then we'd trust each other more, be more creative, more satisfied at work, and so on. This would be good for productivity, so the most ruthless capitalist ought to be in favor of it in the long run.
But I think my comment may not have been clear. I'm imagining something that I hope will be possible in the future. It clearly isn't broadly possible at present.
Also, you should consider that if everyone gets away with being consistently vague, you set a team standard that says people can stay home whenever they feel bad, for whatever reason, and it becomes harder for the company to legally crack down on someone who's doing it for emotional or psychological reasons.