The way I see it, few people are true capital-r Racists but ANYONE—including the most woke and progressive of us—can be guilty of racist actions.
I daresay I am a pretty well-informed person when it comes to issues of race, but I too have had my racist moments. I'm not proud of them, but I'm not afraid of the word "racist" either. I made my racist mistakes, learned from them, and move on.
As you've so eloquently described, there's much handwringing about "racist" being a dirty word, but c'mon, it's not the same as "pedophile." As weird as this sounds, it may be useful to normalize the word "racist" to an extent so that people can get used to the fact that "racist" isn't just about lynchmob KKK types. Anyone can be racist (here I use 'racist' in the broader sense, not just institutional racism).
Most people have done/said/thought something that discriminates on the basis of race at some point in their life.
And it's not a big deal as long as you don't make it a big deal! Learn from it, promise to do better, and become an anti-racist.