I have no experience using a mouthpiece in golf. I do not recognize myself clenching my jaw in golf, but the closest I would come to such an occurrence is on the downswing of a "power" type of shot.
Aside: Perhaps clenching the jaw in golf is akin to grunting in tennis.
However, the article states something I do agree with:
the best swings are typically generated from a relaxed state
To illustrate an example, I was facing a tree on my second shot on a par-5 from the rough. I chose 7-iron as 6-iron was risking hitting my ball against the top of the tree. I hit the shot 200+ yards (I typically hit a 7-iron around 185 yards) over the tree to 20 feet from the hole. My swing thought was to hit the shot solid, and I did.
I illustrated the example above to provide that, based on my experience, a "relaxed" swing is the byproduct of practice, having a concise swing thought, and execution moreso than the clenching, or restriction thereof, of my jaw.