The first thing to disregard is the cushion comment, saying that a nice bushy beard cushions the blow. A beard has very little resistance so it cushions little to nothing, especially taking the recent study of headgear (which resulted in eliminating headgear in olympic boxing)into consideration.
But it obviously does offer improvement and in my opinion because it functions like whiskers. Many animals, both on land and in water use whiskers to sense their soraundings, before it comes in contact with the face/snout.
So If a fighter has a beard that reaches 3-5 cm from his face/chin, this gives him the advantage, of sensing the strikes he didn't see, 3-5cm before they make contact with the face (as opposed to a fighter with no beard, who only senses the strike he couldn't see, when it makes contact with the face), thus giving him the said distance to roll with them, not absorbing them full force. And fighting is a game of inches, as they say.