More alongside the banking system, but we could have governments spend their budget in crypto. They would mint new coins at the beginning of every fiscal year. The coins could pass from department to department being recorded every time they exchanged hands within the government. To exchange the coin for cash, any vendor or individual would have to exchange the coin at a bank while showing identification, and the bank would record the transaction alongside the individuals name and reason for spending the money. The bank would then give the individual cash, and the government would in turn give the bank cash. 100% of government spending would be fully transparent to the taxpayer, they could see which departments received how much money and what they spent it on.

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.