Well, apparently the fourth time is a charm; dug out my old XP laptop and it worked the first time. I followed exactly the same procedure I had been using before (download & install Android sdk v15, download & install JDE, download & install HTC Sync, download Android sdk v13 and extract fastboot). I connected the USB port to the PC, it installed the driver correctly, I pushed the files (TWRP and SU) to the SD card, rebooted it into fastboot from a command prompt, device lost connection and fastboot just showed "fastboot" without the USB part, I unplugged the USB port, plugged it back in, it changed to "fastboot USB" and then I flashed recovery and installed SU. Bang, I'm rooted, just like that.
I'm not sure what the issue was, but I hope I don't have to use my old XP laptop every time I want to flash a ROM...