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Testing is a little different: most professional software engineers consider testing an intrinsic part of software engineering, and so would never say, "that's QA's job". (But even then, there are some especially hairy testing problems that would benefit from a specialist's touch.)
Anyway, I think you're conflating dysfunctional organizations with specialization. In your example, developers say "that's operations' job" because they are lazy and don't care about maintaining a sane production environment. In a functional organization, they'll say the same thing, but they'll mean, "that's operations' job because they will do a better job than me". It sounds the same, but it means something entirely different.
(The same goes for washing dishes. I could wash my own dishes, but I would use a lot of water. If we batch up all the dishes in a 3000-person company each day and wash them in an industrial dishwasher, it will take less aggregate time and use much less water and energy. So while washing your own dishes may be symbolic of teamwork, it's actually a dumb thing to do.)
