This was also covered in Rise of the Vulcans because the original Reagan era extra Constitutional succession procedure was first set up by,

guess who?

Wait for it......

Wait for it......

DONALD "UNKNOWN UNKNOWNS" RUMSFELD; DICK "TURN THE BUNKER LIGHTS LOW AND CALL ME LIEBCHEN" CHENEY AND OLLIE "ABU NIDAL IS MY SECRET MANLOVE" NORTH.

And of course this is serious because, speaking for me and mine, we will not recognize the authority of an unelected executive installed by extra Constitutional means. If he or his/her agents come around here, there's a passel of buckshot a-waitin' fer 'em. Italy has its own hillbillies, doncha know, and that's my stock.

Incidentally, the Constitutional response to a destroyed federal gov't (i.e. the loss of all fed officials listed as presidential successors at one time) is to empower the 50 state governors (or their surviving successors) to take control of the nation. They merely need to elect an executive committee with a CEO to temporarily run the country until they and the state legislatures can set up new national elections. Shit, all they need is enough Congressmen to survive to have quorum to elect a new Speaker and there you go.

The governors are legally empowered executives, elected, subject to the law and the electorate.

Simple. Constitutional. Relatively safe.