The Searchers is weighed down by plenty of racism...
Very little, actually. The behavior of the Indians is easy enough to understand, but for the most part they're seen through the eyes of the white characters and their actions discussed almost exclusively by whites. The whites are bigots - and Wayne does play an unsympathetically and explicitly racist character, which was a departure for him.
Undoubtedly, the film has more nuance where the Indians are concerned compared to earlier John Ford/John Wayne collaborations. In Stagecoach, for example, Geronimo's Indians are a faceless, savage menace who makes war upon whites without motivation. They attack the titular stagecoach because they are evil, and when the Calvary rides in and kills most of the Indians, it is portrayed as nothing less than righteous.
Having said that, I don't think The Searchers is as unproblematic in terms of race as you indicate. What I wrote on the subject when I first saw the movie last year explains it well enough:
Despite the fact that Wayne's character is intended to viewed with scorn by the audience (his sneering in close-ups, and his slaughter of buffalo, drive home this point), the film undermines this scorn by supporting his racist viewpoint. You see, the Indians of this film go around raping women, murdering families, and kidnapping children. The late suggestion that this is retaliation over the murder of Chief Scar's sons does little too alleviate the fact that he and the other Indians are demonstrated again and again to be brutal savages. It's not nearly the revision of the norms of the classical Western that some critics suggest. The Indians are also, of course, terrible shots. Despite being armed with plenty of guns, they can't seem to hit any of the white characters, while the Indians are killed by the dozens.