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Extra credit work: Oh man, now I'm thinking about ways to write a racist and make that racist as likeable as Nate is appearing. Twick, that's a good system for keeping a character's comments in check.
My thought is that he shouldn't be
entirely likeable if he's a racist.
There's no rule that says protagonists have to be entirely likeable, or even likeable at all. I can think of some very good books with protagonists who weren't likeable.
This doesn't mean a racist, sexist or whatever character can't have good qualities, or that their bigoted traits must come from a place of mendacity. There are people who are benignly sexist or racist (they think they're protecting people who aren't quite up to taking care of themselves). There are also people whose sexism or racism (this might go for most of us at some level) are so ingrained, as reflections of the worlds they live in, that they don't even see them in themselves. The old fish not noticing the water they're swimming in thing.
It's not easy, but I've read books where there were deeply flawed characters I still found interesting, even likeable overall, but they also had a trait that was aggravating, even troubling. But since it was part of their arc, and necessary for the story, it worked.
There is a difference between a story where the protagonist's "blind spot" is also the writer's, though, and they are either serving as a spokesperson for the writer in order to push a sexist or racist view, or (in a more subtle way) reflecting a view of the author's that is largely unexamined because it's so much a part of the writer's world that it's like water to fish. We need characters who see their own flaws in this way, but the writer should ideally be aware of them, even if they handle it with great subtlety because those character's views are normalized in their environment.
The latter is one reason why I have trouble reading some of the classics from bygone eras, or at least with getting entirely into them. The various forms of bigotry are so enmeshed that no character ever rebels against them or questions them, or if they do, they capitulate by the end (as in Shakespeare's Taming of the Shrew).
There are modern writers who let their biases show too. For instance, books where every single sexually active woman, no matter how vehemently she insists she doesn't want kids, ends up getting pregnant eventually, or if she can't have children she being deeply sad about it, or else she's portrayed as somehow being a little "off."
There's one writer I really like who does this, and it's a bit of a fly in the ointment for this happily childfree woman who is now past childbearing and isn't consumed with sorrow or regrets.
It's not easy, though, because we don't see our own biases, and who knows what readers from the future might cringe at in our own work, even if we consider ourselves very self aware and enlightened. 20 and early 21st century ideas about capitalism? Meat eating? The way we interact with the environment? Individualism? Keeping animals as pets? Attitudes about the raising of children?
At some point you have to just write what you need to write as mindfully as you can. We can't rise completely above our own biases, but when readers point them out to us, we should consider them at least.