A CP and I got to talking about betas, recently, and it set off a number of thoughts about the process of writing and the process of revising. The analogy of sculpting came up -- the progression from a raw chunk of stone to a rough form (first draft), then a refined form (second draft), and then the small details and polish (third draft and onward, or however many drafts any of these stages require).
The questions that came up were about getting beta reader feedback at various stages, and trying to match up the reader to which stage you were at in the process. That's an issue because writing definitely does involve that can't-see-the-forest-for-the-trees problem -- or, to stick to the sculpting analogy, that you have your nose smack up against the rock.
After some digesting of the question, my gut feeling is that beta readers will crit to the "done"-ness of the manuscript... regardless. If they see macro-scale problems, they'll mention them rather than fine detail problems. I mean, if someone asks me to crit a story where the antagonists seem so clueless that there's no tension, there isn't much point in me giving feedback on narrative voice. If the characters have no distinguishable motivations, that's more important than the dialogue.
The process of fixing the lack of tension, or giving the characters motivations, is bound to change the narrative voice or the dialogue along the way. Everything in a story is connected to everything else. To stick to the sculpture analogy, all those fine details are directly connected to the underlying rough form. You can't have a perfectly sculpted hairdo just hanging in space, after all. (Unless this is a science fiction story...)
When it comes to the refinements of a story -- the narrative voice, the grammar, fine-detail world-building, the particulars of dialogue that make it snap -- I have my strengths and my weaknesses as a beta reader. Everybody does, since they reflect our strengths and weaknesses as writers. At that level, I can see why you'd want to pick and choose your beta readers carefully. Before that, though, I wouldn't worry about it too much.
I suspect that who you hand your early drafts to has more to do with how much you trust them to read that crap and still respect you in the morning. :)
I find I'm more comfortable when I respect my beta readers' writing while still seeing its rough edges. Hopefully, they still respect mine after seeing its wrinkles and bloodstains and the occasional gaping hole.
Do you have your beta readers "labelled" by what they're good at critting for?