Here, in essence, is the problem:

(All that hair is actually wool fibers from the homemade typing mat.)
That piece of cord doesn't extend very far out of frame, and only a few molecules of that classic waxed twine is left on the carriage end. No matter! I have replacement code left over from my last Remington re-stringing, and I see that this has the same, annoyingly-small hole in the side of the drum. On the Monarch, I had to expose the spring to retrieve the old cord's knot and feed in a new knot. Scary, but not impossible.
So I've got the Underwood (née Remington) on its back, and I prepare to undo the back plate of the drum to get at the spring. And here is where Mr. Remington's cleverly-laid trap was sprung! (Every pun intended.)
Against all reason, the spring on this machine is sealed up tight -- the drum is crimped together like a tin can. The screw that I thought held the backplate on actually fastened the whole drum to the machine. The damned thing just fell out in one piece, which is why you're looking at a photo of it sitting on a typing mat, and not a photo of a spring seeing sunlight for the first time in decades.

(What's missing from this photo? See the other photo.)
Clever, Mr. Remington, very clever! Did your repairmen have a big box of new spring-and-drum-and-cord assemblies to fit in, or did your salespersons just try to upsell the next model to the poor
Maybe, just maybe, I could pry the drum apart to get at the remainder of the cord, get a new one in, push the drum together again, and it will all work fine. Or maybe, just maybe, I don't want to suffer. So I'm going to see about finding or fashioning a tiny hook and attaching my new line to that hook, and if that works, just clipping the old line away at the edge of the drum. It's been kicking around that spring for 70-plus years, and it would be shame to break up the set.
[Shakes fist angrily in the air while striking a heroic pose]
I will defeat you, Remington! Do you hear me? Remingtonnnnnnnnnnnnnnnnn!