Subtracting the "people" vs "men" noise, I was trying to draw the distinction between the phrasings "they don't have enough people" and "they have no people left" (in both cases to meaning available to fight).

The former suggests a situation which is quite dire, and that is certainly accurate in regard to Ukraine's current situation. The latter (if taken at face value) is essentially totalistic, and objectively misleading. That doesn't mean that that was their intent, of course. But to my ears it comes across as an overly emotionalized and in any case muddled characterization of the situation.

Kind of like when, say, a startup is going through rough times and someone says "everyone's leaving" when really it was just their friend and a couple other people who have left.

There's a word for this expressive style, btw: "histrionics".