With the USB Drive, devices are controlled by the host device. If what you have them connected to doesn't tell them to do anything, than they won't. For drives like that playing audio would pretty much always be through digital I/O controlled by what it is connected to.
I mention this because really you just need something that is low power that can be fitted into your setup to connect it to to perform that function. The first thing that springs to mind is a Raspberry Pi. It would be possible, I expect, to configure a raspberry Pi to play Audio CDs (or even MP3/etc files on an optical disc) from a connected USB Optical drive, the audio could then output to the DAC/Amp you mentioned.
a Raspberry Pi can run via USB. It will underclock if it doesn't get enough power but playing Audio like this shouldn't be too demanding a task.
If you want to really get into it an Arduino could possibly give an even cleaner solution, but would be more complicated.