I took the leap! Today I removed the TI DAC3203 IC (datasheet here) from the Echo's motherboard and broke out the I2S lines to a cheap eBay PCM5102A DAC (datasheet here) board. Aaaaand after a little fiddling with the DAC... success! It works beautifully. Demo video below.

To-Do:

- Try out a couple other better DACs such as the Sainsmart HI-FI DAC meant for a Raspberry Pi

- Test implementation of an I2S to S/PDIF conversion IC (this one) to hook the Echo Dot up to a digital home theater receiver or optical input DAC.

- Re-instate the Echo's internal speaker, and somehow make it less clunky to switch between using the internal speaker and the audio output. I'd love to be able to somehow detect the status of the amplifier (hooked up/powered on or off) and switch the outputs with a microcontroller, but it seems like the best solution as of now is to voice-control the switch using an esp8266 configured as an alexa "smart home" device. Looking for suggestions here.

- manufacture a board to add a digital output, and 3D-print a larger volume bottom shell to house the additional board plus the internal speaker.

- ???