[Eliot Ness]

We're not out of the woods quite yet, though. I let the radio play for about 30 minutes, and when I went back into the room after breakfast, I was met with a very powerful smell. Burning plastic or something. It was still playing fine, but the whole room reeked of that burning smell

I'm on the fence about replacing those wax capacitors...........

Dave, so something is overheating in your radio and you are on the fence about replacing the highest failure items in a radio of this vintage? 

The only way you'll know if those wax caps are good is to test them for capacitance and leakage.  And even if a couple of them test OK today, that's no guarantee that those 75 year old caps won't go bad the next week, next month, or the next time you turn it on.

Capacitors are cheap and easy to replace, and like Mike said, if you're concerned about looks just restuff them so it looks original.  It's your radio and you can do what you want with it, but if you're going to play it safely you need to find out what is overheating and why.