This would be awesome if the HQ augment were assigned with a piece's original purpose in mind. We've seen on Empyrean armor that the current team actually understands what equipment does and when it does those things, so maybe they would actually do this.
Even if not, it couldn't make the useless pieces more useless or the single-job-ability pieces less useful, so why not?
I also like that it would give the +1's a more palpable benefit.
The only issue is that the pieces couldn't be stored or mailed if this were done. I guess not augmenting the worthless pieces would solve this issue for most folks, though.