I think the problem with blackface isn't a white person appearing as a black person per se.

The problem is that (a) blackface has a sordid history of being used to represent gross racial stereotypes of blacks as portrayed in minstrel shows, not actual blacks, and (b) blackface typically implies a stylized appearance (e.g. burnt cork, or shoe polish) conforming to those stereotypes, rather than any attempt for the performer to actually look like a black person.

Contrast with Key's performance, or other similar contemporary performances (Eddie Murphy is a classic example), in which the purpose is not to celebrate stereotypical appearances of white people (although usually there are some mild stereotypes involved, usually with regards to speech patterns), but to derive humor from either the concept of passing as white, or simply to give a black entertainer the opportunity to play a white role. For instnace, in the above clip, the message of the clip relies in no small part on the police officer being white (yes, there are minority police officers who are also trigger-happy against blacks, but there's not much room for such nuance in a clip like this).

Also, there's many more cases of Key and Peele portraying more stereotypical black characters in their show than stereotypical white characters.