> whether the biasing of the output stage affects the total output power.
Insignificant. (For push-pull fixed-bias.)
Try it yourself. Bias-down as far as you can, say 5mA. Around 1 Watt out you may get horrid wave-shapes as each tube "wakes up" from its "too cold" bias. But up near full power the peaks will be fine. And you will get very nearly the same maximum unclipped power whether biased cold or hot.
OTOH----
> biased to (for example) 20 watts, then that 20 watts also represents the maximum signal output of each tube?
This is true for *class A* amps. Input power is constant. Signal diverts power from tube to load. Efficiency (on sine wave) is 50% at best. One 6V6 at 12 Watts DC can make about 5.7 Watts of clean sine output. Two EL84 P-P with cathode resistor bias makes about 18 Watts (violates the 50% efficiency because we often let them run hotter than 12W/plate, push-pull keeps distortion lower, bias-shift happens, and the EL84 does not hard-clip so "18 Watts" may be significant rounding).
However quad 5881/EL34 amps are most-often "fixed" bias, idle cool, and the exact idle does not affect maximum output.
> output transformer was replaced
Loading is critical to getting rated power. I have a 12 Volt tractor and I put a 12 Ohm lamp on it. I get 12 Watts of power drain. Now I replace the bulb with a 24 Ohm job, I only get 6 Watts. The tractor will hold 12V despite a few lamps; tubes are not so solid. Too-low loads also reduce power.
As a rough guide, quad big tubes run about 2K plate-to-plate; however unusual plate or screen voltages will shift this optimum. Post your voltages and expected power output.
There is not a ton of difference between 100W and 50W. If the tubes are old and abused, this may be your entire problem.
> measure the maximum size of the signal on the grids
If you have a DC-coupled 'scope, put the trace on the top line, poke the grid. Adjust gain so the grid's negative DC bias voltage puts the trace mid-screen. Apply signal until lightly clipped at the output. If the peak grid voltage almost touches the top line, and is otherwise clean and symmetric, the driver is doing fine. If it has to touch the top line to get "maximum" at the output, the power tube has no margin and is weak.
If the amplifier has NFB, as you exceed clipping you will get very wonky wave shapes. The NFB is trying to "force" the power stage to give more than it can give. Interesting, but not relevant if diagnosing low power output.