... my understanding was that the OT contributes to distortion as it approaches saturation, as it begins to clip the waveform, essentially limiting the voltage swing in order to supply the current demanded. is this accurate?
That's what I thought a long time ago. Now I'm not as-sure we run a transformer to a point it materially distorts the audio.
Regardless, we tend to learn the wrong terminology from Amp & Pedal controls, where "Drive" and "Saturation" may be labels on controls that allow more/less signal to pass. And if you turn those up, something else in the circuit winds up distorting.
On the other end of the spectrum, transformer designers say "saturation" to mean that any further increase of voltage or current
cannot cause the transformer's magnetization to increase. Even if we
can apply enough/to-little signal to a transformer to encounter distortion in the transformer's transfer-curve, we haven't "saturated" the transformer the way designers mean.
If we do "saturate" a transformer (or inductor), it stops being a transformer. Inductance drops to zero and the couple-hundred ohms of winding resistance is all we have present at the transformer primary.
Ohm's Law ---> Voltage = Current x Resistance
Power = Voltage x Current
Resistance ---> kilohms fall to hundred-ohm = Primary Volts falls very-low
Saturation Power ---> Few-Volts x Same-Current = Watts drop to almost-nothing
So when you saturate your output transformer for-real, speaker volume drops to almost-nothing. Most likely, you heard your amp very-loud when you heard the distortion & just assumed the output transformer was "saturating/distorting" along with the output tubes (and/or preamp).