Why not?
If you're trusting a central authority anyway, then all this is unnecessary, and you can replace it all with an API over a database, as paulmd's post suggests. But then you've eliminated the whole reason why people use cryptocurrency, and what differentiates it from Visa.
Under such a system - how do you know that there is anything behind the government's numbers? They could be pure fiction right now and no one but the government would know. Also, the government takes a cut. As the whole economy flows through their currency, they become a de jure taxing body. Their taxes drain the economy, and can be used to manipulate the economy to the government's whim.
It may shock you to learn that not everybody trusts the U.S. government. The growth of cryptocurrencies worldwide is largely fueled by the fact that the set of people who don't trust the U.S. government is larger than the set of people who do - and in fact, they've seen their biggest up-take in societies like Zimbabwe and Venezuela where trust in civil monetary authority has completely collapsed.
I am no economist, but I have never thought the idea of "trust" had much to do with the value of US currency. Certainly it's not backed by gold any longer. But I think it's backed by something real. Bombs. The US has more than enough to force others to accept their currency as having value.
