One day, people were using commodities like rice and salt as money. Even after creation of fiat, people were still comparing the papers we had with commodities such as gold.
After some time, governments started to print those papers and made the fiat lose the value gradually.
The dirty fiat needs to be replaced by a new type of currency. But its takes time.
Yes all fiat Will go in stable coins.
There isn't a big difference between fiat and stable coins. Both are controlled by a third party. 
