Stick, I think you are confusing induction motors with the universal motors usually used in these type of saws.
A universal motor will slow down on low voltage. That is one( not the only, nor the most common, perhaps) way of speed control.
Extension cords will indeed heat from high current devices. A 15 Amp saw is indeed a high current device!
An induction motor, such as an air compressor usually has, may indeed not even start if too long, or too small of an extension cord is used.
The thyristors(triacs) used in most speed control/soft start circuits operate as switches. There is near zero voltage across them when on and virtually no current for the time they are off. Either results in very low power consumption in the thyrister itself. This is why they can operate with very, very small heatsinks! By the way, they turn on and off 120 times per second(on60Hz), not just when you turn the tool on!
So... will a long extension cord be a problem? You will indeed have less power available to the tool. With a universal motor that may cause heating because it is pushed harder to get the job done... however it should run all day long without a load with no problems. Dull blades make this much worse with or without undervoltage!
Now with in induction motor? Whole different story! It will indeed burn up with too low voltage applied.
Remember snow melts above 32deg F or zero deg C. An extension cord really does not need to get very warm to melt snow. I wouldn't worry too much about that in and of itself. The allowable temp for the wire is much higher than that!
A bigger issue to me is ground adapters. Don't use them on high current devices, period. This include microwave, air conditioners, dehumidifiers, etc. Most power tools don't run long enough to be as big a problem, but I have seen many, many, many
of those thing just melt form prolonged high current... remember you have two pair of relatively high resistance connections in a very small space in those things
I have actually not seen many(can't actually recall any) speed control modules fail from heat per se. All though it probably does speed up capacitor failure somewhat. The bigger issue is the quality of components used. Replace the failed one with a high quality( and properly rated!) one and you will likely not see a repeat failure. Loose connections from vibration is another common failure.This can of course cause arcing and burning issues, but is a symptom not the root cause.
So all in all I agree that we should limit the length of extension cord as much as possible. Also use the proper gauge cord for the tool being used. And I absolutely agree that it should not be used coiled up. Won't attempt to explain why except to say it will indeed heat more that way!
I am probably rambling a bit, and YMMV, but that is my $.02 worth!