I purchased two replacement burners.
One task which could have gone either way is that on this stove, the burners mount onto trim rings. It was a challenge to remove the old burners from the rings without bending the sixty year-old rings, but I was ultimately able to do it.
The alternative would have been to buy new trim rings, but it's unlikely that the replacements would have exactly matched the look (and aged patina) of the original ones.
Important safety note - route the electric wires through the ceramic holder's channel, to the side. Don't just go out the back, as wear over time could damage the wires if routed improperly.
All told, this work took about twenty minutes per burner, most of which was fussing with the trim ring.
The old stove is running on all burners, and the vintage look of the kitchen is still there. Thanks to AppliancePartsPros for stocking the replacement parts for this project!