Blackface has been a pressing concern for black people for centuries. Blackface was a key component of the minstrel shows of the early 19th Century, in which white performers would play slaves and free blacks. These portrayals would be built around insulting, degrading, and dehumanising stereotypes of black people: the aggressive "buck" with his lustful eye on white women; the freed slave who aspired to polite society but couldn't pronounce his words correctly; the unkempt, neglected black children who were disposable even to their own parents. Blackface demeaned black people, and desensitised white audiences on both sides of the Atlantic to the horrors of slavery.
How relevant is that to today? I don't know but many black people still find blackface to be offensive. Would the joke have worked just as well without the blackface? Almost certainly.