I think the point is less about Black's voice and more about his acting range. There's no requirement that a character has to sound a certain way. Charles Xavier was a New York native in the comics, but they still cast an Englishman in the role. And if you insist on the Thing having a deep, gravelly voice, then it could be processed to sound deeper, like they did with Ruffalo's voice in Thor: Ragnarok.
His acting range in the Tropic Thunder clips does not look any different to the rest of his career. Almost every scene has something in it that feels exactly like his other performances with the sole difference being that this particular character isn't as inoffensive as usual (but that's really just down to the facts of the character, not the performance).