There are too many different failure modes to attempt to generalize. Each failure mode may or may not take other components out when it happens. This depends on the failure mode _and_ the particularly circuit topology the tube is being used in.
By failure mode, an example might be: Internal arcing means that 2 parts of the tube have high voltage felt across them. If one part fails mechanically and the parts get too close to each other, then you'll be seeing sparking/arcing and likely a number of small flakes of metal being thrown off. You can inspect the tube and see them rolling around.
Another failure mode is where the filament wires or the support fails and they fall onto other parts of the tube containing high power. Given that these 2 parts of the tube are operated off of separate circuits in many power amps, the chances for adjacent component failure can be high, again, depending on how the circuits are wired.
So, to the question of whether it will fail more at startup or during normal operation after warming up, the proper answer is: It depends on the failure mode.
In the case of the more contemporary Mac tube amps, though, most all of the power supplied to the tube's various circuits are all regulated if not computer controlled. This implies that you cannot really predict the exact failed components until a tube fails.
For this reason, I like to have a 'panic' switch set up for my old tube gear where I quickly ensure that the all AC power can be removed in a hurry. It works, too, for old SS gear, too.
Cheers,
David
