Adafruit TLC5947 breakout board is not working with my Arduino Uno
... the LED does not turn on when connected to channel 0 on the board. [...] Connecting the LED directly to the v+ line lights it up, but obviously isn't switchable.


From this I suspect that you have connected the LED between pin 0 and ground. The problem with doing this is that the TLC594x LED driver outputs are current sinks, and connect the output to ground when on. The LEDs must connect between V+ and the output as shown in the datasheet.