Dividing by zero is more an inconvenience (or frustration) to me because whatever calculation or program I was working on comes to a screeching halt. Give me an opportunity to change the last thing I put in, but don't suddenly screw up the whole result just because of an error on my part. Changing the result to NaN (not a number) isn't much better. The idea of "nullity" or whatever it is that supposedly has an answer to dividing by zero sounds more like a way for systems to continue working when that happens (just so they don't have to handle it themselves), and yet you'll still be locked into a predetermined answer (in this case, nullity) that will probably screw up the rest of your calculation.
As for why dividing by zero leads to death and destruction... well, some systems have failed before because dividing by zero made them crash and they neglected to check for that. If the universe divided by zero and forgot to check for that, then you'd probably see the destruction they're talking about. Even if the destruction is usually a black hole.
