Treasury bonds set the lower limit for interest rates in the currency. Why would anyone write a 30 year mortgage at 3% when they can buy a bond backed by the go

Treasury bonds set the lower limit for interest rates in the currency. Why would anyone write a 30 year mortgage at 3% when they can buy a bond backed by the government for 5%? The central bank wants people to be able to get loans so they prop up the price of the bond.