Maintaining drip pans on coil oven hobs
I missed your point about heat completely until you pointed it out.

If you have a textbook-perfect black-body of a drip pan (totally non-reflective, thus not shiny), it will radiate heat in all directions once itself is heated by the coil above it.

If you have a perfect reflector of a drip pan, it would not absorb any heat and simply reflect any heat back up.

A silvery tray is much nearer to being a perfect reflector (still a long way off) than a shiny black one to a perfect black body.

I have no measurements or data of any kind, just as an educated guess, I would lean towards having a shiny silver tray rather than a shiny black one if you are concerned about efficiency. Besides, a silvery foil tray is most likely going to get less hot itself, less heat absorbed. However I am questioning how much of a difference is actually there between the two. I could be wrong on this without any data, but I suspect that would not be your main heat loss, convective losses around the coil (spaces under and along its circumference) and your sauce pan and radiative losses would far outnumber that.

You said they are non-shiny black metal trays. Could they be coated with a catalyst to self-clean any food residues like you have inside self-cleaning ovens?

Edit: you might want to check with the manufacturer about using shiny liners just in case if the coils are not designed to cope with reflected heat.