This is an explanation I can gather from understanding their in-game behavior (focused only on their key traits expressed in combat). It may not be exactly accurate as this is how I interpret it. The concept of Shan't is based on a trait which shows that Cool Mii characters are too stoic to pay attention to their friends. Notice how the other quirks benefit themselves in a show-offish way. Cool Mii characters are basically those overly dramatic, smug people that love to show off, with the lack of proper regards to other people as a setback (which is why Shan't doesn't have the potential to start quarrels, no reason to be angry when the Mii in danger doesn't expect themselves to be covered while at the same time the Cool Mii chooses to stoically ignore).
Laid-backs may be lazy, but they're not heavily dramatic (Get Serious plays around the character trope of "don't mess with this seemingly lazy character"). Stubborns are heavily determined but can be reckless (shown by Bluff, as this is expressed by the Mii insisting they can take care of themselves). None of which go with the concept of Shan't.