PS - No offense intended to our native american friends!
Fortunately, we have a screen shot from the video. I think it gives you the basic idea.
I trust I don't have to explain what's wrong with this.
Who posted the video?
The phrase "unfortunate intersection" suggests that the people who danced aren't the ones who posted the video. Perhaps the posters were bystanders who happened to come across the dancing family and filmed them. Or perhaps the family posted the video on another site and someone reposted it.
If the people who posted it weren't the dancing family, that sheds a different light on the "no offense" comment. The posters may have posted the video to illustrate the racism and stereotyping in our culture, not to revel in its ridiculousness.
This is the reason Native Appropriations and I often post "offensive" material: to expose our society's prejudice against Indians. If that was the posters' intent, then no apology is necessary.
For more on the subject, see Racist "Make Me Indian" App and What's Wrong with No Doubt's Video.