No idea about what's raised in the standard literature.
For practical purposes, the key question here is not how important the non-brain body is to cognition, but how much of the variation among individuals is explained by differences in bodies vs. differences in brains.
If the variation due to body differences is sufficiently negligible, then the "personality extraction" process can be done with just the brain, and uploaded into a standard body (whether biological or mechanical or digital or energetic or what-have-you), and the resulting person will be a "close enough" match to the original.
If not, then the resulting person won't be a close enough match.
Personally, I suspect that most people vastly overestimate the precision-of-duplication required to be a close enough match for practical purposes. I suspect the differences in cognition caused by the loss of original somatic context will be swamped by, for example, the differences caused by the original social context.
Of course, if one wants to drop the practical stance and instead ask metaphysical questions about whether it's "really the same person" without the original body, then different issues are salient... of course, that's true even with the original body. I have trouble seeing the point of that, though.
Incidentally, it seems to follow from this line of reasoning that quite a lot of the brain is unnecessary, also. Though the state of the art in neuroscience is nowhere near up to being able to freeze just the necessary bits.
It leads to an interesting thought experiment, though... if I can only preserve enough information to allow an N%-match copy of me to be constructed at the other end, at what point does my intuition that "I" have been "brought back to life" (or teleported, depending on the particular intuition pump being used) collap...
Many approaches to cryonics assume that a detailed map of the neural patterns in a brain (via brain scanning technology) may be sufficient, using future technology, to bring that person "back to life." But cognition is greatly shaped by more than just the neural pattern: it is shaped by biology - by the body. (See Noe 2009; Pfeifer et al. 2006; Lakoff & Johnson 1999.)
I admit I'm pretty unfamiliar with the cryonics literature. I assume this is a standard objection, and has standard responses. Where can I find those responses?
Thanks!