I'm afraid that could lead to political instability. Maybe not, but I imagine if 59% of people vote "X" but 60% were needed, people could revolt or at least drastic and unpredictable changes in voting in the next elections could happen - "how can this political regime ignore the voice of the majority?!".

You'd need most of the people to understand why 60 or 66.(6)% of people are needed to decide something and really believe in this threshold. And Y% of the populace is different psychologically than Y% of elected officials (in cases where a supermajority of officials are needed to pass Z in a forum like parliament/house/senate).