Yeah, bad wording. 338*2 is in the range, but is nowhere near the 12 tones we use when A=440 (and wont be with integral doubling or halving). The property "within the range A4=440 A5=880" doesn't have any particular special musical meaning.

The idea I wanted to convey is that 440-880 is just a convention, starting from 440 as concept pitch, but concept pitch can be totally different (and outside the range). 432 is a famous standard which has and is been used today, but also tons of others, even widely outside the range...

(also: "but it's a perfectly fine not" --> "note").