This is called the Federal Reserve Bank and it is indeed separate from the tax and fraud portions of the federal government in the US.
But the rest of your post is pretty silly. How does a secure digital exchange protect against fraud? It doesn't.
Fraud can occur when a party lies about assets it owns when entering into an agreement, which would be prevented if you had to present proof of ownership for any asset you claim to own? Most ponzi schemes would never happen.
You can own something, accept payment for it, and never actually relinquish ownership. Besides, the blockchain is only proof of ownership of things on the chain. Real world things are not covered at all.
Right now, we have the government operating as both the "miners" and the auditors.
A middle-ground, then, might be one in which there is a general-purpose public blockchain where the government takes the role of the trusted third-party on-chain, thereby lowering the overall societal costs of things like unfulfilled contracts while still ensuring that my oil-reserves are what I say they are.
From the government's perspective I see this as desirable but falling prey to the principal-agent problem. We'd need virtually no civil-court judges or attorneys to prosecute white-collar crime anymore. It'd lower costs, but in doing so unemploy many of the people in the system.