They're very welcome to, but it will only hasten free-market crypto's rise.
Many governments have raised the idea of their own cryptocurrency, a "Central Bank Digital Currency" (CBDC), but to date none of them have come to much.
Consider the following problems a government has when launching its own cryptocurrency:
- Legitimising competition: Governments launching their own cryptocurrency legitimizes the idea of private cryptocurrencies to their entire citizenry, and helps to popularise them as alternatives to government fiat or digital currency.
- Misaligned incentives: A government is undoubtedly going to build in taxation controls, spending limits, privacy invading tracking, lack of interoperability with other cryptocurrencies or other anti-consumer "features" that most or all of their citizens will not want, and which free market cryptocurrencies do not have.
- Brutal competiton: The cryptocurrency market is absolutely cut-throat and exploding with innovation. A CBDC is going to be swept aside in a tide of decentralised productivity. In the same way that no single government could make a better national Internet that outcompeted the global Internet, no country will be able to make a cryptocurrency that can innovate, improve and attract as much human talent as the large free market cryptocurrencies can, let alone the combined global pool of private market cryptocurrencies.
- Other Priorities: Similar to trying to ban cryptocurrencies, governments are way too busy with all the other things that need their attention to fight this battle. Creating and running a cryptocurrency is no small task, let alone all the surrounding infrastructure (wallets, marketing, payment systems, browser plugins etc. etc.), and it would be a huge drain on government time, money and attention that would still be inferior to just allowing private cryptocurrencies to flourish - which is free for the government to do.
See also: What if governments ban/regulate crypto?