It would be enormously useful to governments and central banks and I think we are headed to a cashless economy. I think blockchain will be used to record a lot of things for data analytics. For central banks, a cashless economy would allow for negative interest rates and remove a lot of expenses involved with printing and handling physical cash and coin. Like you said it would also make it impossible to use central bank note in a black market without leaving a record.
Difficult to be realized. Imagine for a moment that internet don't work. What it will be with the digital money. Total chaos. No one can buy nothing. People in need to buy things. Banks in need to work. Federal Reserve in need to distribute the needed cash for the everyday needed. etc. etc. And the last thing. An economy cannot survive without monetary policies and with the digital cash this is impossible to be realized.