UPDATE WITH CARNAGE PICTURES
Finally dove into it today. Pulled the receptacle out and wired up a household receptacle. No love. Opened up the junction box at the slide. Everything was tight and looked good. Popped the cover of the fixed end of the short cord that goes to the slide.... And found this disaster!
It appears the wiring for the receptacle got a bit warm. Given the condition of the wires for the microwave, which were mostly intact, I'd say the wire nuts weren't properly tightened down, which caused resistance in the connection. We often use a small space heater on that receptacle (or at least used to), and that likely caused the heat in the connector that eventually melted-burned the hot wire in that stranded piece of flexible cord.
I pulled both sets of connections apart. Although the cords are the size of 10 gauge wire, they're rated 14. I picked up a new piece of 12 gauge cord at lowes for a buck a foot. I'll make sure the wire nuts are actually tight, and that there's sufficient contact between the stranded and solid wires.
Related: is it normal to use wire nuts to connect stranded cables to solid wires?