I got it done, thanks, everyone.
The threaded part was in about 6-7 inches and jammed. The drill-spin did not work but twisted up the cable up at the pedal end, making it harder to work with. I clamped a vicegrip to it and hit it with a hammer for a while and that did nothing. So I took out my 1/4-inch socket extension and jammed it in the other end. It hit metal about two inches from bottoming in the hole.
So I hit that with a hammer and the cable bit pushed through pretty quickly.
Putting the new one is was fun too. Could not get the threaded part past the burr or bend that held the old one tight. So I tinned the cable about five inches back of it and cut it off. The new cable went in easy without it, and these cables are a good 16 inches longer than they need to be (the old one was looped and clamped).
So the new one will be cut and clamped. No biggie. I got a few extra clamps for good measure.
I also cut about 3/4 inch off the tube so the bowden will have close to the right bend after moving the transaxle forward an inch and some. Sag was over 2.5 inches with the stock pipe, and it's the shortest bowden they make. I got a little better bend using the old bracket, and the 3/4 inches off the cable tube got the rest of the way.
So we're good on clutch.
Now working on the throttle cable.