OK. Let's break it all the way down explicitly.

The joke is that Chang's character is in Drow makeup, and because Shirley is out of touch with games, she thinks that the Drow make up is blackface.

The depiction of the Drow makeup has chosen an "ambigious" shade of black/grey that made it seem to some viewers or people who want to see it that way , as blackface, in real life, not as a character in the show. This caused problems.

The suggestion, is that if they chose grey or some other ashy shade , and or gone more elaborate with the make up, like red contacts, or pointy ears, they could have done EXACT joke, but made it not "ambigious" and avoided the heat that the episode has actually gotten.

I would even say that having a LESS blackface looking make up would further accentuate Shirley's out of touchness, making it funnier, as it would be more ridiculous that A) She would think that make up is blackface and B) Shirley thinks that's what Chang thinks making himself look like a black person would entail, red eyes and pointy ears and a non human shade of darkness.

Furthermore, expanding on my other point which was not adressed in your post :

I guess I’m out of touch with the race politics jokes of 1890s, so the parody seems out of time for me.
Maybe if they threw in a tin pan alley, player piano or travelling curio joke, I’d have gotten it a bit better.

What the fuck are it's always sunny et al doing making fun of this outdated shit in the first place? Did they get that mad at Spike Lee making Bamboozled and using blackface as an analogy of people acting up to stereotypes to get ahead in life , that they had to address it as a commentary on blackface, an "artform" from the 1890s as a target?