Bad title. Should be "Why didn't they have cutting edges?" As the still picture shows, that polearm had a sharp point and there are plenty of suits of armor that have nice, square holes where they were punched through.
I made dozens of arms from 1100-1500 and owned two historic pieces, one English polearm and one Turkish mace. None of the heavy ones had cutting edges. All of the lighter swords did. Don't be fooled, the edge on that poleaxe would cut you clean if swung and connected with flesh. It just wasn't made to "cut" armor. Armor isn't like sign sheet steel, it's thickness depends on its location. The breast would be twice the thickness as the sides for instance. It was also tempered, not soft and would dull a sharp blade quickly.
Cutting edges returned when firearms made heavy armor obsolete.