Is there any fossil evidence that beards appeared at the same time as we stood up, got opposable thumbs, and started fist-fighting? It sounds like thumbs are "for" fist-fighting. In order for a trait to be selective, your reproductive fitness needs to be influenced by it? Jaw size has been shown to be favored by women (during ovulation). Having a heavy jaw must be more protective than a bit of stubble, no? What about clothing? First off, when did we start wearing it?
Crazy thought: clothes? I would argue that the most important atvantage afforded by clothing is not to keep out the cold. Erect posture? It allows us to see danger/prey at a distance, and to throw rocks/spears at it, but it also allows us to CARRY things. And that's also the advantage or our thumbs. Again, semiotic/macromemetic analysis, people! With erect posture and thumbs, and with the addition or wearing clothing everywhere (even when it's bloody hot), we have a sack handy that we can fill up with food windfall and then CARRY it for miles back to our village without overheating. We're not hunter-gatherers, we're hunter-gather-CARRYERS.
Again, macromemetic analysis. What changes if we have this feature? What changes if we have this behavior (or artifact), i.e., clothing, even in very hot climes. Anyway, it's just a theory, but it's a theory that brings with it a lot fewer assumption, and needs a lot less supporting data than that we have beards to keep our faces warm or to protect us from a light slap.