I'll ask the obvious: wouldn't the aircraft just take to the skies directly, without bothering with the formality of setting their transponder, if they were knowingly escaping an apocalypse scenario?
Plus transponders are really convenient when you’re trying not to crash into other air traffic. Particularly in a scenario where you might be expecting ATC to be unavailable or abandoning their posts.
If governments and airspace control have already collapsed, post tense, then of course anything goes.
Worst case scenario a fighter jet will be scrambled to investigate.
But in apocalypse scenario, chances are the fighter jets will be busy with tasks other than enforcing FAA rules.
Depending on the type of event, they very well could be scrambling to shoot down unidentified aircraft.
Fog of war sucks, and friendly fire still happens often.