The concept of doing the video wasn't the problem. Doing a video to communicate "the plan" would have been fine. The problem was choosing Eugene Melnyk as the person to deliver the message. (And I can only assume that it was Eugene Melnyk who "chose" Eugene Melnyk.)
After everything that had happened in the last 12 months - from the comments at the outdoor game to the awkward team photo to all the rumors and innuendo about Melnyk and Karlsson - any reasonable person would have told you that putting Melnyk in the video was a dumb idea. There was no upside. It should have been Dorion, or Boucher, or anyone else involved in hockey ops. But Melnyk did it anyway.
So with this Twitter thing, sure, you're right. A reasonable person would say that there'd be no upside for Melnyk to carry it out. But that certainly doesn't mean he didn't do it, based on some of his decisions in the past.
You're making the assumption that Melnyk is a reasonable, logical person in all situations who understands what he needs to do to, and what he shouldn't do, to positively shape public perception. But we know that this isn't true.
Like
@coladin said, there was no upside or logical reason for Bryan Colangelo and his wife to do it... but they did.