Like the modern Chinese, most Native Americans did not grow much facial hair to begin with. What sparse facial hair did grow was typically plucked out as soon as it appeared, according to accounts written by whites who lived with or near them. They didn't shave. Modern Native Americans often have a bit of admixture with Europeans or their descendants and so are able to grow a bit more facial hair, or at least don't mind letting it grow. So you may see them with mustaches and beards. In China, too, you can sometimes see men with relatively sparse beards and mustaches. But the really big and bushy beards and mustaches are sprouted by men from the Near East, Europe, or Africa. Or else the men from these areas really do shave, mostly with metal blades -- something Native Americans didn't have before Columbus, as they used stone and bone as their primary resources for cutting tools.
In South America the men used the sharp teeth of a small rodent to cut their head air in a sort of bob, according to anthropologist Napoleon Chagnon. But even there, Chagnon says they plucked beard hairs as these came in. As a woman past the age of menopause who occasionally grows a beard-like hair, I can attest to the fact that one can simply grasp it with the fingernails and pluck it out, as long as one doesn't mind the teensy-weensy pain that causes. If one is going to talk about tweezers, one is going to allow the Native Americans to develop metallurgy, which opens a whole can of worms. Some of those around Central America had a little bit, but most didn't, you see. So what was everybody else doing? I tell you, just as Daniel Boone and all those other occasional captives tell us, they were just plucking them out as they came in. It's really not that big a deal when there aren't that many of them. Besides, after you've done it the first time, the skin toughens up and it's not that hard after that.