Besides, I don't think Shakespeare was picturing Othello as an African. I seem to remember reading somewhere that the writing of the play was prompted by the arrival at the English court of ambassadors from the Ottoman Empire- not Africans but Middle Easterners in huge turbans. If Shakespeare is drawing on pre-existing stereotypes it's the medieval stereotype of the Saracen- the Moorish Knight of the Mummers' plays. He's not thinking Morgan Freeman, he's thinking Omar Sharif.
Shakespeare had his hang-ups and prejudices but they aren't ours. He predates most of the things we feel so guilty about. His world view is still very largely the medieval one with a few holes punched in it- Eurocentric with a vague awareness of Africa and America and India being out there somewhere (does he ever even mention China or Japan?) We can read the Holocaust into The Merchant of Venice and Western Imperialism into The Tempest (the plays will carry almost any weight we choose to impose on them) but we also need to remember- in justice to the man- that these things hadn't actually happened at the time of writing.