Yeah, the issue seems to be that, when an iPhone loads the page containing the video, PBS does some user agent checking to determine what video source to provide.

When iPhone goes to AirPlay, it provides the AppleTV with the only stream source it received.

Ideally, either the HTML5 video tag could define multiple streams based on size (similar to how it can provide different streams based on video format), but since that's not in the spec, I suppose we have to conjure up a "standardized" convention.