It has some flaws in the system -- it essentially creates a new government minted currency and the banks would need an incentive to be party to this transaction. It solves the problem of lost government funds, and helps create a paper trail that anyone can audit to detect embattlement faster. It might impact the government's ability to fund some classified operations. You can't exactly say "I am {First Name}, {Last Name} and I am cashing my paycheck for working at the CIA" to the entire world.
This would have no impact whatsoever on the fraud in this article.
The person claimed they were spending money on approved purchases and provided documentation to support it. The government then gave her the money.
Adding crypto would change nothing, other than making it more convoluted.