what happens when a transformer is very heavily loaded?

[FoMoCo]

If you were to gate the 2X rated load at 50% duty cycle, everything would be quite ordinary.

Not true.

Copper loss is due to the square of current, therefore twice the current is four times the loss. Factor in your 50% duty cycle and copper loss is double.

This is because the rms of a pulsed waveform is not the same as the average. You effectively calculated average current, which is not what is causing heating in the winding.

 

[Jsixis]

I have seen some commercial "undersized" supplies described as being constructed that way to provide a lot of "dynamic headroom" for short durations. For instance early NAD amps.

and I had a set of speakers that died to that "soft clipping" NAD design of a 2200 amp 🙁

 

[CharlieLaub]

and I had a set of speakers that died to that "soft clipping" NAD design of a 2200 amp 🙁

The soft clipping circuit has nothing to do with the power supply. It's done at the input stage:
http://sound.westhost.com/articles/soft-clip.htm

Some NAD and Proton amplifiers like the NAD 2200 had 6dB or more of headroom. This means that high power can be delivered for brief (<100ms) bursts, but longer periods of demand will cause the rails to sag enough that continuous power is one-quarter or peak power.