All -
I have never noticed a burning smell in a belt-drive, not unless something is wrong.
I try to monitor my machines as best I can, and from experience with that I can say for certainty, even when my 1986 Kenmore 70 was new and in my Mother's 1982 70 before that when it was new, it IS very common for the power cords to be warm from use, but they should not be hot. This past weekend the cord on my 1980 built machine was warm to touch, and it softened the insulation a bit and made the cord more flexible, but it was by no means hot. These machines can draw up to 840 watts when under load (spin acceleration is the most taxing) so the cord is going to warm up during that period.
There are a lot of reasons why an older machine may make some funky smells when running, but if the source is truly electrical, I would be concerned.
One thing I noticed with the machines I'm using that have been stored long-term in my storage trailer - it has a wood floor and a distinct smell inside. When I run a machine that was housed in that trailer (one or two were in it more than 10 years) you can smell the trailer when the machine warms up. This is odd because there is nothing in the machine except metal and plastics and I wouldn't think they would retain old odors, but clearly they do. Some of these machines I have completely cleaned, disinfected, and de-funkified, yet they still generate a faint amount of that woodsy smell from the trailer.
Randy - as to your machine, your noisy wig-wag may be producing some electrical scents when in spin since you're moving a conductive metal plunger inside a polarized electric field which is making that bizarre sound. You may also want to be sure the motor isn't heating up 40-some years of dust, belt-grime, and bug or rodent carcasses. Check the motor for overall heat - it should be warm, but not difficult to keep your hand on immediately after a load. It could need lubrication. The centerpost bearings could be hot also (take the agitator out after the final spin and feel the centerpost of the basket up by the drive block - if it's hot you need to lubricate the bearings).
Water dripping on a hot part can also create strange smells. One of my machines when I took it apart had a spider's nest and rodent droppings and seed shells everywhere.
It is never good for a Kenmore/WP to stop because the thermal element in the motor kicked it off. David - if you have a smell of wires burning, stop using the machine until you know what the issue is. The motor may have worn bearings, there may be drag in the pump, the belt could be stiff, or all the above.
I've seen this said before on AW.org as mentioned by others, but we should all be careful in assuming that we can take a vintage machine, and press it into service as it would have 30-plus years ago, and expect it to not have 'issues'. Fourteen loads in a week is a lot to ask of an un-tested or un-serviced machine. At least in my opinion.
But, after all this above, other than a warm power cord, your belt-drives should not be making 'scents' of any kind other than your laundry additives. Take a shop vac or a dust buster vac and clean-up cobwebs and lint build-up. Wipe away any oil deposits on the gearcase, and make sure wiring connections are plugged-in tightly.
Gordon