If we're talking about the minstrel show-inspired American/Western tradition, performing in blackface has always been racist. I honestly don't know enough about historical uses of blackface like Shakespeare's Othello to give much of an answer in that regard. However, the song-and-dance routine in blackface is pretty firmly rooted in the minstrelsy tradition.

That strain of blackface has always trafficked in stereotypes and caricatures of black people. In the antebellum US, it spread the pernicious idea that slaves were happy-go-lucky buffoons, granting cover to the institution of slavery. They even co-opted anti-slavery messages like the one in Uncle Tom's Cabin, and are the main reason why a generally sympathetic character from that book who dies defying slaveowners is now used as an epithet for subservient blacks.

Now, I know you're probably more focused on the "unacceptable" part of your question, and that depends on where you are. In the US, and specifically Hollywood, Abraham would have been around the time of the last hurrahs of blackface. The musical show and film This Is The Army also featured a blackface musical number and was released a year afterwards. But attitudes had changed enough that by the late 1940s, blackface was not a viable thing to do in something you'd want to mass-market.

Notably, even though the Amos N Andy radio show was quite popular, and based off the same stereotyping as blackface minstrelsy, the 1950s TV show used black actors rather than the original white voice actors in blackface. Even then, it was protested by the NAACP and cancelled within a couple years.

Now, outside the US, things were a little different. In the UK, the BBC broadcast the Black and White Minstrel Show until 1978, and the show continued on stage for another 10 years. And in Australia, there was a controversy in 2009 over a blackface Jackson Five tribute group that got shown on TV. They had actually played that same TV show 20 years prior and presumably escaped controversy back then enough to get re-booked, giving you a sense of attitudes towards blackface may have been in the 80s in Australia.

Blackface has continued to be used in media even in the US, although not in the minstrel context. Robert Downey Jr. in Tropic Thunder and Gene Wilder in Silver Streak both did blackface in a winking way for comedic effect, and mostly escaped criticism for doing so. And many opera productions of Othello, including those at the Met in New York, have used blackface in recent times. But using it in the way you describe has been seen as generally unacceptable in the US since the mid-1900s.