First I know from reputable reporting that nature programs do this all the time.

Second, I consider the chances that the elements of the shot lined up naturally. The owl is swimming, an incredibly rare event. Its doing so in front of a documentary crew. Its doing so in waist high water so its easy for someone to closely follow it with a hand held camera. Its doing so during the three or so hour period that the lighting is perfect in the slot canyon. Its doing so in a part of the canyon that starts out of sight of land then rounds a blind corner to reveal safety. Thats a lot of low probability events that need to happen for what we see to come together.

Third, I look at the condition of the owl. Swimming looks very taxing for owls but when he gets out of the water he doesn't look bedraggled and exhausted. In fact he jumps around to dry himself then turns to look at the camera. He's also very clean for a creature supposedly struggling in muddy water for a considerable time. All in all he looks like a bird that has been swimming exactly as long as he have been watching him swim.

So really I don't know that its staged but I'd give to astronomically high odds that it was.