...because first of all, WAR, yuck, the big one, 20th century wars of annihilation; awful, no thank you, also the Naziness of it all.
Secondly, it's inaccurate for what it means to say. The word "blitz" in the warfare context refers to the blitzkrieg, a strategy where (do not correct me on this) the Nazis go faster than their opponents. They pass the front and overwhelm the enemy forces, and you cannot effectively respond, because they're on blitz time and you're not. That is... not what's happening in blitz chess, where you're both going fast. You're both just playing chess, but fast style.
The Blitz, the German bombing campaign that people in the UK called "The Blitz," and which (I think?) brought the slang "blitz" into common use, was not really blitzkrieg at all; the British people apparently just called it that by vernacular association with prior blitzkriegs known to have been carried out by the Nazis in Poland and France.
I know the naming of these things was not planned, and prescribing corrections is equally as silly, but just to be clear about the history: There were already people playing fast chess, and some calling it "lightning chess," as far back as 1897. "Blitz" means "lightning" in German, by the way. But people didn't call this category of time controls "blitz" until the 1960s, by which time the primary connotation of the word, worldwide, would assuredly be the form of warfare we have been discussing, not simply lightning. So, that's kind of a coincidence. Lightning is fast. Makes sense. The name "blitz chess" may be inaccurate about what it means to say, but it ends up being accurate about what it didn't know it was sort of saying, etymologically speaking. HOWEVER:
"Bullet chess" (which is a category of time controls faster than blitz chess) was not a thing until much later, like the mid 1990s maybe. Some guy just decided to call it Bullet Chess, probably because Whizz-O Chense was taken. The thing is, lightning is faster than bullets!!! Like, way, way faster. I'm not going to crunch the numbers on this. It's some tens of thousands of times faster than any given bullet. This is an error on the scale of mistaking car for boat.
So, Blitz Chess is named after blitzing generally, which comes from "The Blitz," named erroneously after something, the blitzkrieg, where a lot of people were killed using bullets and other bullet-shaped things, which was named after lighting, which is faster than bullets, but gives name to the chess mode that's slower than bullet chess.
Don't even get me started on "rapid" chess, which is only played with rapidity for some of the game.
An obvious solution: Horses. The horsey piece was right there in front of us the whole time. Horses famously have different speeds. You know, the horse speeds. Walking Chess (you can take a walk and think about your next move)... Canter Chess (fast)... Galloping Chess (faster). We're skipping trot because it really does not roll off the tongue. I think this is good. I think I'm gonna sleep great tonight.
EDIT: I feel like I missed an opportunity here to point out that when "Blitz Chess" already existed, named ostensibly after a bullet-shooting thing, there came to be an effort to name a new and distinct time control for chess, and the name they landed on was also about bullets.