I think that by withholding the speaker's race until the last line, Cullen is able to conceal the main point of the poem and thereby make it more of a surprise.
As a reader goes through the poem, not knowing how it ends, he or she is likely to wonder where the poem is going. It seems more like a reflection on the ways of the world, and about why God allows what the poet sees as injustice. But then when one reads the last lines, one is struck dramatically by the fact that the poem is about race.
If the poet were not black, it would sound pretty patronizing and perhaps even racist. It would sound like why would God make some black person (with the implication that they're inferior) a poet?