Yes, in general, it would be better but not necessarily easier. To start, the same angles would be best if included in any video but the problem with video is that perspective must be maintained and it's difficult to keep the entire head in perspective while holding a phone at arms length going through all the motions. A tripod would obviously help with this but in my experience, the majority of video sent to me when I worked in clinics (and even now) wasn't great because the perspective wasn't considered. Doctors can't make an evaluation if the video is too close or the angle of view keeps tiling one way or the other while the phone is being moved about. I've never once had a video sent to me that was shot on a tripod.
Lighting is also important and the rule of thumb I discussed about the position of the light source is usually more of a problem because it is more difficult to shoot steady and usable video in the washroom like I was showing for photos. To do it right, and to make the video as accurate as possible, there is more involved when compared to taking photos.
Another problem with video is file size. If you're shooting one long continuous video it may be too big to send through email and then you're looking at having to use services like Google Drive or Dropbox to send files. Easy enough to do but then you're asking that the clinic be tech savvy enough to handle this type of transfer (you'd be surprised how many are not).
Ultimately, video is better if it's done right especially now that most new phones shoot 4K video. In fact, with 4k video you can take screenshots of the video to use as photos as they are roughly equivalent to about a 12mp photo which is more than enough (in the right light) but this video and infographic are designed for the most common denominator scenario and for those that have zero idea of how to do this right, which is most people. Kudos to you for using video and obviously well enough to get an assessment. If the assessment was accurate and you got what was estimated, and it turned out the way it was supposed to. It means you did it right.
But, I do plan on making a video about how to shoot video for proper assessment and documentation. I can't include everything at once because no one wants to watch a 30 minute video covering it all:)