Well, I can't really mark this as "Solved" because with each of the possible solutions, I've made some headway, and gotten good partial results on some systems, but not on others. However, it occurs to me that there IS one VERY easy way on just about any system to record the output as input.
It's a hardware solution:

(picture of Y splitter cable - headphone jack style. 2m 1f)
You have to use headphones (or you could use your speakers, on a desktop) to monitor it, since plugging one of the ends into the headphone port will kill your speakers, but it's not as bad as one would fear, and it's worked on quite literally 100% of the systems I've tried it on, which is better than our track record for any of the software solutions.
These things are available at your local Diode Hut (Radioshack) for about 4 bucks. Plug one end into mic (or line-in) and the other end into your headphone jack. Use the female port for a pair of headphones, and you're 100% likely to record what the computer is playing, provided you have your mic (or line-in) port selected as the record source.
This also assumes your headphone/line-out port works.