For a given damage value x, blunt and pierce will always do more damage than cut, regardless of armor. Armor cannot mitigate damage from blunt or pierce sources as well as it can mitigate damage from cut sources. Thus, a 30b or 30p damage weapon will always deal more damage than a 30c weapon, whether the opponent has 30 armor or 60 armor. 60 armor just reduces the damage of a 30c weapon more than it reduces the damage of a 30b or 30p weapon, compared to 30 armor.
However, sources with cut damage typically have higher damage values. A more realistic scenario is comparing a 36c weapon with a 30p or 30b weapon. The 36c weapon will end up doing more damage against lower armor values, while the 30p or 30b weapon will end up doing more damage against higher armor values.
For weapons that have the equivalent damage value, cut weapons typically have additional bonuses which promote their use: faster speed, longer reach, and/or special properties such as bonus against shields.