It works! Yay! The main problem all along was that I was running the board at 5 volts, not the 3.3 volts it needs. It was able to respond to JTAG commands at the higher voltage, but it correctly reported that it couldn’t write its flash memory under those conditions.
I still need to update the README for the JTAG Whisperer, but the code in the repository should work right now. Hooray!