The thing about battle axes is that you don't need to cut trough plate either, the weight behind them can cause a lot of blunt trauma as well (if heavy enough and not as effectively as say, a mace), knocking enemies out of combat. With an axe you have all the weight on the tip, again, same as using a long sword griping it from the blade, you change the center of weight. It all goes back to the same point, the damage the Polish guys in the video take is not that different from the real deal, and non blunt weapons, agains the armor they are wearing, wouldn't have had a better effect. In fact, one of them in the video actually used a poleaxe, he could effectively have killed someone with that, however their tactics are more aimed at winning the game inside the "pit" they are in than actually killing someone. As a result their moves are not something you would see at HEMA, they are more brutal and less precise, if you used a sharp weapon to stab someone trough a slot or weak point in the armor that would be "illegal", so it would be that all of them had poleaxes and their hits were aimed at vital areas I would think. EDIT And no, the axe as a weapon of war was not around a lot in the late middle ages, like Beaushizzle said. A viking versus a kingth in full plate would have been a figth from different eras. Eventually everyone adopted the newest things the others were using, so is the arms race.