I got it.
See, when you said that a cowstail can be a QAS, I thought you meant that one can wear it clipped to an ascender. This is how I wear my long cowstail, with it clipped to a Basic. And as you and others have described, I use it as an upper ascender, as a QAS, and as a cowstail (by clipping the biner to a loop while letting the Basic hang).
On the other hand, what you were saying is that a cowstail without an ascender is a sort of QAS, if the rigging allows. That I understand.
Some people have told me that what I have is not a QAS. Whatever.