Well... I don't think you can nail down language, especially idiomatic phrases, to that level of specificity.
"Can you X" literally/etymologically means "are you able." If you were to translate, you'd translate to "will you." I don't think the etymology is coincidence, or lost on an average speaker. "Can you" feels softer and more polite. It reflects something about how people want to interact with each other. You have to leave work early because your mom "couldn't" watch your kids, not because she refused to.
Incidentally, in Ireland we do say "will you," "would you" and "are you able to" more commonly than "can you," which sounds slightly american/international to my ear. Oddly (or not), "will you" is (IMO, locally) more informal. You'd say it to friends, when making trivial (pass the salt) requests. "Would you not" is also (I think) an irish choice of words. It's used to make suggestions, rather than requests.
These things don't bother us until/unless they're coopted into a different context, and used tactically. Going back to the original point about corporate-drone speak.... The "can't" vs "won't" language is used in the first person to obscure responsibility. I didn't call it Orwellian because it's evil or onerous. I called it Orwellian because it affects culture/thought deeply. The language helps maintain an impenetrable ambiguity, implying that every contentious decision is actually not a decision. It's dictated by regulators, or at least by "corporate."
Also, in context... it's an (passive?) aggressive way to cut off a conversation. The equivalent of "Good day Sir!"