"Why is it that many whites can only support a cause when a white face is attached to it?"

This should be rephrased.  It should say:

"Why is it that people struggle to accept those that do not visibly look like them, and trend to lumping like with like?"

Go to any major city and you'll find little enclaves where people of a certain stripe have decided to live near each other, run businesses to support their little mini community, etc.  We always feel more comfortable surrounding ourselves with the familiar, rather than the 'other'. Black, brown, white, yellow, red ... doesn't matter, we all do it.

But that's precisely why 'cultural appropriation' should happen.  Keep people to their enclave, and only let whites make white things or blacks make black things or Asians make Asian things, and you'll rapidly find that their customers silo, too.  It's our natural human tendency to do so, and we have to consciously work across the grain. The wonder is in the mixing.