There is one overriding reason for biasing output tubes. Every tube operates in three ranges, depending on how much grid voltage it is seeing. At the very low end, there is a range where the current through the tube does not vary linearly with the grid voltage. You obviously do not want the tube running in this range while playing music. At the very high end, there is a range where the tube also becomes nonlinear...it cannot deliver current linearly with the grid voltage any more because it is at its capacity. In between, the tube operates linearly...as you vary the grid voltage the current flow through the tube (and hence through its part of the output stage) varies linearly. This is the useful operating range of the tube for our purposes.
The bias presets the grid voltage so that with no signal applied, the tube is operating at the very bottom of its linear range. Set it too low and you will get distortion from the tube operating nonlinearly. Set it too high and you will be running more current through the tube at idle...and thus generating more heat, wasting electricity, and prematurely aging the cathode of the tube...with no benefit. That's why the actual bias setting is so critical.
Now, as in all things, tubes are not absolutely perfect, and neither are they all the same. The transition from the nonlinear to linear ranges are not precisely defined, and even the "linear" range may not be truly linear, with minor variances occurring. Often there will be regions near the low and high ends of the "linear" range, where variations occur. A particular tube may need a bit more idle current, and hence bias, to minimize distortion, than another; and generally if bias is set a bit higher than what is often "recommended", the distortion WOULD go down. In the shop I used to set bias and then do a distortion measurement at low power, then tweak the adjustment to see if the distortion would be reduced. Sometimes it was surprising how this kind of critical adjustment could actually make a difference. Other times, by doing this, I could detect tubes that were mismatched, poorly made, or nearing the end of their useful lives; in a lot of old tubes, even if they still sounded OK and "biased properly" the cathodes would be near-depleted and this would show up in distortion measurements. Granted...not everyone can do this. I can't anymore, myself, because the test bench setup had been meticulously tweaked for low noise and no ground loop hum; now that I've retired, my garage test bench is nowhere near as quiet and I sold the ultra-sensitive distortion analyzer, anyway.
I'm not meaning to scare anyone off. Dynaco allowed for this when specifying the Biaset procedure for the Mullard tubes it included with new amplifiers, and in fact most times I could set the bias on stock Dynaco amps lower than their specified 1.56 volts (whatever idle current that REALLY corresponded to would depend on which amp, and on the value of the cathode resistor). Other tubes, as noted, vary in their performance, and in their bias requirements. Worst of all: tubes with poor quality control, whose bias requirements vary from one tube to the next. On some of them, the internal element spacing can vary so much that a particular tube may HAVE to be biased much higher before it operates linearly. The best way an individual user can minimize this problem is to buy from a reliable vendor and have the tubes matched, in pairs or, in one of those huge amplifiers using four output tubes for one channel, quads.