However, if we choose to ignore this for a moment, there's a good counter-argument: the way to get rid of debt is to pay it back, and the way to pay it back is to work more and invest wisely. The individual or household that needs to pay off some debt would do better if, say, the people in the household got jobs (if they don't have them) or extra hours at work (if they do). Allowing the unemployment rate to rise as it has done in countries operating "austerity" doesn't fit that analogy at all.
My point is not that the above argument is correct, just that it's possible to argue both for and against austerity using "intuitive" arguments based on analogies with households, and these are rarely correct. That said, neither are most of our mainstream economic theories! (At least some of them must be dead wrong, as they can't all be right).