In the early days of MMA I also saw some wrestlers use it to good advantage. As Matt says, it's useful for someone who hasn't yet developed a strong boxing foundation to keep from getting knocked out while they close the gap to grappling range. I still teach a variation to beginning jiu-jitsu students for that purpose. It takes less time to train a beginner to some sort of basic functionality with this simple approach than with a standard boxing guard.
You don't see it much in modern MMA at the professional level because the sport has progressed so far. A skilled fighter who can box, kick, and grapple will take advantage of the openings the guard leaves. Also, a pro fighter should have enough boxing skill to not need this simplified approach.
Rodney King of Crazy Monkey Boxing developed a similar approach for a different reason. His foundation is in boxing, but he observed that the average beginning student (one without natural talent, who didn't grow up boxing or brawling) would freak out in sparring primarily due to the fear of getting hit in the face. Either their technique would fall apart, or they would quit training, or they would stay in training and take a lot of head trauma by the time they developed good boxing skills. By starting them out with a structure which is very good at protecting the head (although it has other weaknesses), these students were better able to relax, focus on technique, and not develop brain damage before they got good at boxing.
