If I understand correctly, anything with the VC getting overly hot and potentially damaged should also be accompanied by the smell of the adhesive burning... Is this correct?
"Yes a voice coil getting overly hot", will cause a smell if it melts something. Wire is made to smell a certain way just for that purpose. You identify problems BY SMELL.
If I understand your question correctly.
I don't know if you can, BUT pull the driver and inspect the interior of the enclosure. IF wires are dangling, I usually silicone and tape the loose wires in place. Inspect the driver and SMELL it. If it has damage it usually shows signs of bluing or discoloration also. IF you inspect the driver and it passes a visual inspection with care taken to LOOK at any foreign materials that may be stuck to the magnets or between speaker components, give it a shake. Adhesives that fall off behind dust covers are a pain. I've had a few of those. Shake the driver and listen. If it's behind the dust cap you can hear it. Either type of phase plug (fixed or not) or dust cap can be removed, to remove debris inside the cavity.
It's always a good idea to rotate drivers 180 degrees every couple of years. The bigger the driver the more it will become part of normal maintenance. I have Jensen Imperials. They get the treatment every 3 years. They were my FIL, it was actually part of the original instructions when he commissioned them to be built in the late 50s. Rotate the drivers, check securements and check electrical connections. They are over 70 years old now, they look new. The surrounds are pleated silk. 100 years or more maybe? They will outlast me.
Get a stethoscope keep the volume low and listen. I have found a multitude of issues with a scope, including bad plywood panels with voids causing weird sounds.
Last but not least. throw that recording away.
Happy hunting.
