No, because it doesn't work just as well, because everyone comes up with different words.

The NATO alphabet isn't just used by NATO, it's used worldwide by civil and commercial aviation personnel (pilots & ATC). There's a reason for this: these words were carefully selected because they're unique and won't be confused with other words.

It absolutely should be taught in school; it's not hard at all, and everyone has problems spelling words out on noisy communication channels (like cellphones).