Questions on making a minimal/custom Arduino circuit @ 3.3v/8Mhz?

I'm sure that what you are trying to do is possible. It should not matter whether you use a 8MHz crystal or 8MHz internal clock, the SD card does not have an internal time reference so it doesn't care what the exact clock speed is. I suggest:

  1. With the SD card removed, run some sketches on your board to test it out. Include sketches that toggle the pins that you use to communicate with the SD card.

  2. Check your connections between the mcu and the SD socket carefully, looking for open/short circuits.