Tanks for your concern and your tips, fellow members of this, the greatest forum on planet Lever!
Solemn words for this the solemnest of solemn moments: the operation of our electric patient was successful, no more does she smoke or smell. Espresso is extracted with great ease and joy, for miss C is now warm and cosy, neither too hot, nor too cold. Behold: Her heat element is emitting heat, her bimetal strip is expanding and contracting as the boiler water's temperature rises and falls, and her little micro switch is clicking on and off with.... well, not meticulous timing, but at least it is clicking on and off.
I must say it is a great relief, so much easier to make espresso when you don't have to keep boiling water and pouring out lukewarm water and rinsing the group to heat it 2-3 minutes before EVERY shot.
I had a closer look under the hood today, and saw that my heater support, a strip of metal that I had quickly cut to size out of some old sooty BBQ grill piece of iron and put in there as a replacer for the original heater support which I had forgotten at work when taking the machine home over the Christmas weekend... phew, where did the end of that sentence go? Anyway, this DIY support was causing at least half of the smoke. It seems some of the soot still had some flame left in it, and was ignited now and then, causing occasional wisps of smoke. Now I know why chimney sweeps are needed

Took a trip to the office and found the original support. Smoke gone! And so is the burnt Teflon/circuit board smell, which Allon correctly described. So some of the smoke was from the burner, but it is broken in now.