Vertical output transistor breakdown.
Would that be likely a failure mode of the transistor itself, or of whatever is driving the transistor?
Failure mode of the vertical transistor. I've seen this failure many times in old CROs. I admit, that not Tektronix, but mostly the vertical transistors are heavily stressed. High voltage (that may have become even higher due to many reasons), heat dissipation, etc.
And due to manufacturing defects or due to the old high voltage transistor technology not being perfect, the breakdown voltage may decrease with time, until it fails completely.
Problem is, that checking such transistor is not easy, as the failure mode is not easily detectable, unless testing in circuit. So get the relevant schematics of that scope, and get another scope to probe the input signals to the vertical output stage and probe the very output of that stage.