I'm able to AirPlay a video from the Roku Channel website via Safari browser on iphone or ipad to any compatible streaming device, Roku or non-Roku, like Apple TV device. To do so, I just swipe down the Notifications screen from the top (ipad or iphone) and then tap the AirDrop/AirPlay icon from Safari's Now Playing control. I have a screenshot showing proof, but I'm unable to upload it yet here. I have now uploaded screenshots successfully.
from an iPhonefrom an iPad
I still wonder why the Roku mobile app omits AirPlay or Chromecast features from its own video player. The Roku website apparently hasn't done that yet.