Ziga - I've seen many discussions and comments about how a beard improved a fighters chin. These are my thoughts on it based on what I know.
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.
There are two key flaws to your theory.
FIrst, animal whiskers are not the same as facial hair. Technically, these are specialized sensory organs called vibrissae; stiffer than most hair fibers, more highly ennervated (making them more sensitive) and commonly with some degree of independent muscle control - they can be used like a blind man uses a cane.
Second, if we allow that there is a indeed 3-5 cm buffer, that allows a fighter to sense a strike, we need to consider that this translates to, for a strike with a reasonable velocity of 6 m/s (Olympic boxers throw at +/- 9 m/s - http://bjsportmed.com/content/39/10/710.full ) provides an advance warning of 0.8 milliseconds.
Reaction times to tactile stimuli are on the order of 100 ms (http://www.jneurosci.org/content/26/42/10879.full).
Binocular processing (that is, seeing the punch coming) provides a lot more warning time; predicting from body motions even more. In other words, experience >> beard.