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?
If they have 5 minutes, sure. If they have 5 hours, they'll follow procedure.
Colliding with other planes is going to impede your escape plan, so it would still be a good idea to turn the thing on. No further action needs to be taken for the ADS-B output to be correct, it works once it's powered on.
In a theoretical scenario of the billionaire class of the world having some kind of "advance warning" of the apocalypse, they'd be taking to the air in the hours or several days prior to a total disaster happening. Meaning this would be done while the local governments were ostensibly still functioning, in which case you can't just have your private jet depart without active ADS-B and in-the-clear voice traffic for ground, and air traffic control coordination.
If governments and airspace control have already collapsed, post tense, then of course anything goes.
AFAIK the transponder kind of turns itself on when powering on the plane, you'd have to explicitly disable it but then you'd have weird discussions with the airport tower guiding you to a free timeslot on the runway which would just delay your takeoff, since ignoring the airport tower is a good way to not get off the ground at all because you'll accidentally be hit by some other plane.