I think it's a matter of taste. I just watched your link and it seems performative to me. I imagine a video from one of his earliest performances would have come off as authentic. But decades later, does he really even feel the same way as when he wrote it?
As someone who has performed a number of songs (some of which I wrote) over several decades, and has played one cover song in particular at least several hundred times... every performance is unique.
It is a different experience for the performer than for the audience. As an audience member, I get tired of any particular song very quickly. As a performer, it often feels like a new challenge each time. One night I can struggle to feel anything at all and it's like I'm phoning it in. The next afternoon I can come away from a spontaneous run-through of the same piece and feel like I've found new insights and played it better than ever before.
That being said, while I recognize it's somewhat subjective, I vastly prefer Jeff Buckley's version of this song over all others. Maybe because I fell in love with that whole Grace album at a pivotal point in my life. If I'm going to pick a song I feel like Cohen fully owns, it's gotta be Everybody Knows.