Transistors are switches, and mainly amplify current. Audio transistors are designed to have a long and fairly even switching pattern from full off to full on to allow for volume changes in amplification - at least those in vintage audio equipment. Class D is different, so this explanation isn't useful there.
What happens in a complimentary or quasi complimentary system is that the transistors on each "side" (plus and minus) are operated with some minimum current flow -- in the mA range -- for a couple of reasons. First, it takes time to switch a transistor on from zero, and secondarily it is necessary to have enough current flow through the complimentary pair at the output will transition from positive sine wave to negative sine wave froms without a "step" as the slower transistor (NPN I think, but I'm dylexic and confuse them all the time) switches on. Manufacturers specify this bias current, usually as mV across the emitter resistors of the output. It can also be set with a sine wave generator and an oscilloscope, which is probably better.
What that means when a rail goes off (blown fuse or something similar) is that the pair is STILL conducting, only with one side at ground potential, hence with no connection between the speaker plus and minus (ground) terminals, rail voltage will "leak" through at very low current flow. The speaker protection circuit will open the speaker relay or ground the input to the amplifier to shut off current flow as DC at any significant level and current will certainly overheat voice coils. The drivers and outputs with functioning rail will still have 1V or so on the base and will conduct, but at low current flow. Which is good, else they would fry.
Center voltage (or offset) is set by the differential pair resistors at the first stage in the amplifier. Some amps are adjustable, some are not and rely on carefully matched transistors in the differential pair for offset control. Offset adjustments will have no effect on an amp with one rail open,
Note that output transistors and usually the drivers are connected between the rail and the output, usually with only emitter resistors in the circuit. The emitter of the driver will be connected to the base of the output transistor. With both rails hot, the potential of the center of the output emitter resistors should be zero volts with some mA level current flowing from the positive to the negative rail (or actually negative to positive since that's the actual electron path I think). Take away one rail voltage, and the current flow will be zero, but the opposite rail voltage will appear at the center of the resistors!