Why would a blower motor work with A/C, but burn out with furnace

Okay, so I think I figured out the reason, and I learned a lot about HVAC in the process.

The answer is that a ceiling fan is moving air at basically zero "static pressure." Static pressure in an HVAC context means the amount of resistance that the air has to moving. In a free environment, that's zero, or close to it, but in a tightly restricted system of ductwork, it will be much higher. A ceiling fan is rated at such-and-such CFMs... but at zero static pressure. They don't tell you this because it's obvious; you're probably not mounting your ceiling fan in the opening of a four-foot-diameter cylindrical duct. If you found a curve of CFM versus static pressure for any given fan, it would probably drop off steeply. But of course there would be no reason to because the whole point of a ceiling fan is not to push air through a duct, but to circulate it in an open room, so the motor is optimized for that use case.

This is compounded by the fact that most ductwork is terribly done, so furnace manufacturers err on the side of oversizing the blower to make sure the unit isn't starved of air (causing it to operate poorly or even die early), or incapable of supplying the living environment with the required heat. Either of these things would make people blame the furnace and make the furnace company look bad.

So the short answer is that you have to use a powerful blower in an HVAC system because ducts suck. A ductless unit/system could probably get away with a much less powerful blower because it wouldn't have to work nearly so hard.

I'm going to suspect poorly insulated ducts running in unconditioned space as a first guess. I don't know the layout of your system, but it's very common to take duct runs up into the attic, and even if they are wrapped with duct wrap (R3-5 typically) that's abysmal. If the ducts are poorly wrapped or not wrapped in spots, it gets worse. R-50+ is far more appropriate for hot air in a cold attic.

As for the furnace exhaust, it should be fairly warm - otherise you get to visit the series of questions that are on here about furnace exhaust (high-efficiency condensing) freezing up in low temperatures and shutting the furnace down.

Fundamentally, if your furnace is running 24/7 and the temperature continues to drop, it's undersized for the load - but one of the best ways to deal with that is to reduce its load by improving insulation and air-sealing (caulking things - the most boring, most bang-for-the-buck way to improve efficiency in most houses that have not already had that done). You could also add a second furnace to take some of the load, but then you'll be burning even more gas. You do not want a furnace that is much larger than your "design load" (ideally, your furnace runs 24/7 and maintains temperature on your coldest days) since an oversized furnace is inefficient .vs. one that's just the right size.