Firstly, wow! You really went above and beyond give me a thoughtful and comprehensive response, so thank you very much. Secondly, I meant to post this in the quantum forum, as that is what I am studying at the moment. Of course having understood the majority of your post, I also see that this will hold for the quantum case where it is continuity of probability current that is being being described by the equations.
So, if we were to make a measurement that violated the continuity equation, could we also say that gauss' divergence theorem has failed? Or could the discrepency arise from somewhere else? I guess this is one of those 'If unicorns existed, what colour would they be?' kind of question, so I'm not sure if it is really a valid question, but I thought I'd put it out there.