For instance, in Warband, the Iron War Axe has a cutting side and a spiked side, but it only does cutting damage? Wouldn’t it be cool if you could switch between the spiked side and the cutting axe side, thus switching between piercing and cutting damage. Maybe it could lead to balance issues?
This is not a balance issue, but a gameplay issue.
If there is a weapon that can do 30 cutting damage and 12 piercing damage if you press mode switch, then no one would switch mode because it is not worth wasting precious seconds required in competitive duels and melee fights for a very slight armor piercing property. There is no point. If such a feature is implemented it requires it to be seamless and easy to transition, which is not possible in a game like Mount and Blade unless there is a sort of "Smart" System that can automatically detect and determine which types of moveset to use when attack, but that brings its own set of problems that would be very disliked by the competitive community, which is inconsistent damage because of the Smart System, and the very real chance that false positives can occur such as lose you a kill because it switched to piercing while you attack a naked man because an armored opponent appeared next to him.
A very popular suggestion to fix this problem would be to
"Nerf cutting damage so piercing can pierce armor, like real life!"
Which brings up the same problem as the "two different types of arrows for bows" problem we have had one week ago.
This is a point where fun>realism has presence. If you have the choice between being able to kill everyone or only being allowed to lill a select half of people, them people always pick the first option, as limiting your targets is a stupid choice in a competitive play and renders many equipment useless.