I guess it depends on your definition of evidence. An online AA publication claims he was born in Kenya, and I have read reports that his relatives make the same claim. I have seen no official documentation of any of this, although FWIW I have seen rumors that a Kenyan birth cert exists.
My suspicion has always been that this is not about where he was born but about him being an illegitimate child. The original birth cert would show that.
Would anyone care today? I'm about as right-wing as they come and I consider it comprehensively irrelevant. Certainly very darn few of his present supporters would abandon him because of illegitimacy.
Perhaps some of his present opponents would care, but nobody else.