Overheating/insulation destruction of Transformers
When a transformer overheats through overcurrent, the copper windings heat first. Eventually, the wire will reach a temperature at which the insulation begins to degrade, to lose mechanical strength, to smoke and char, to catch fire, to melt (not necessarily in that order).

If the heating occurs very slowly, say a 1.5x rated current overload, which will eventually give a 2.25x rated temperature rise, then the transformer iron will be at a similar temperature. A temperature sensor, or cutout switch, mounted on the transformer iron, would be able to trip, and disconnect the power, without the windings being much hotter than the switch.

If the heating occurs rapidly, say the 20x current you might get from shorting a transformer, which will give you 400x the power dissipation in the windings, the windings might reach a dangerous temperature for the insulation in seconds. There will be no time for the core to heat from the windings, so a thermal switch on the transformer would be useless. Fortunately a current of this magnitude will easily blow a fuse. 

This difference leads to the standard method of protecting a transformer against all overload scenarios, a fuse of 2 or 3 times the rated current to protect from short circuits, time delay so it doesn't nuisance break on transients, plus a thermal switch mounted on the transformer to disconnect on lower overloads. The thermal switch will also protect against high ambient temperatures pushing the transformer temperature up as well, should cooling fail for some reason.


  What have you actually witnessed?


A long time ago I used a (quite expensive) signal generator from a well-known manufacturer. Internally, the power supply was 'crowbar' protected against over-voltage. If the over-voltage detector tripped, it was designed to trigger an SCR directly across the power supply output, to blow the primary instrument fuse. Unfortunately there was too much residual resistance in the circuit (or maybe somebody had replaced the fuse with a higher rating (don't do that)) and the fuse didn't blow. The transformer cooked, boiled off insulation, and condensed grot coated every surface and switch contact within the instrument, rendering it beyond economic repair. We noticed the plume of smoke and unplugged quickly, averting a likely fire.