As Kris had mentioned, the time investment required to make your own FM transmitter that can read USB/SD card would be incredible...
There are multiple file systems that could be present on the USB drive or Card... there are multiple file formats, and bit-rates...
From here, you have two choices... cross your fingers and poke and prob at random spots on that main board and maybe perhaps a support component is faulty... Now that power is fixed the only idea I really have, is checking the IC for correct power... From there, it's all up to the mystery chip under that black goop which is not repairable.