Some things you got wrong there. There is not less metal in an axe than there is in a sword. Axes are cheaper, yes, but it's because they are much easier to make, not because of metal costs (well, that, too, axe metal doesn't have to be the best steel you can get). Also, axes don't cut deeper than swords. This generalization is wrong. An axe is basically a sharp wedge, so it's good for splitting semi-hard materials like wood. However, the wedge form also means it's decelerated faster in soft materials. The real reason you don't chop wood with a sword though is because of tip energy. An axe concentrates the mass in the tip of the weapon (hence unbalanced), which, given it is accelerated to similar speeds as a sword, yields much higher kinetic energy. So yes, it is potentially more damaging than a sword slash. Which is already somewhat represented in the mod.
Which means you are basically right, axes work better against armor than swords, but they don't penetrate it, either. What they do is break bones under the armor, IF you can get a straight hit (which is hard to do on curved plate).