Can a person's race be included in a statement without the statement being "racist"?

You have it backwards. “African-American” is the term specifically used to describe black Americans who are descendants of slaves. “Black” is the more generic term to describe anyone with African ancestry.

And as for the rest of your post, it sounds cute but from my experience those who want to pretend that race doesn’t exist usually claim so only because it gives them an excuse to not acknowledge racism as it exists.

I’m not accusing you of being racist, just telling you that your perspective (assuming it is genuine) won’t sound so noble to some ears.