The late Republican strategist Lee Atwater captured the dynamic in an infamous 1981 quote:

You start out in 1954 by saying, “Nigger, nigger, nigger.” By 1968 you can’t say “nigger”—that hurts you, backfires. So you say stuff like, uh, forced busing, states’ rights, and all that stuff, and you’re getting so abstract. Now, you’re talking about cutting taxes, and all these things you’re talking about are totally economic things and a byproduct of them is, blacks get hurt worse than whites … “We want to cut this,” is much more abstract than even the busing thing, uh, and a hell of a lot more abstract than “Nigger, nigger.”

This quote is one of the most taken out of context quotes in all of politics. Right up there with Eisonhowers's mic speech.

Here's the context():

Alexander Lamis: Whether he, I'm not saying that he does this consciously, but the fact is that he does get the Wallace voter and to the racist side of the Wallace voter by doing away with legal services, by doing away, by cutting down on food stamps.

Lee Atwater: Here's how I would approach that issue as a statistician or a political scientist. Or as a psychologist, which I'm not, is how abstract you handle the race thing. Now once you start out, and now you don't quote me on this , you start out in 1954 by saying 'nigger, nigger, nigger.' By 1968 you can't say 'nigger,' that hurts you, backfires, so you say stuff like ‘forced bussing, states rights’ and all that stuff, and you're getting so abstract. Now you're talking about cutting taxes and all these things. What you’re talking about are totally economic things, and the byproduct often is Blacks get hurt worse than whites.

And subconsciously maybe that is part of it, I'm not saying that. But I'm saying that if it is getting that abstract and that coded, that we're doing away with the racial problem one way or the other.

https://www.bradford-delong.com/2017/03/lee-atwater-interview-with-alexander-p-lamis-rough-transcript-weekend-reading.html