Blockchain ensures transparency, so you have to ask yourself if governments are interested in more transparent public systems, and what would be the advantages for them in adopting Blockchain technology on these matters.
Personally, I don't see great points for governments to implement Blockchain on administrative, educational, health areas, because such data is considered confidential in most cases. Moreover, it would leave trails, in case members of the government tried to fabric or manipulate data at some point, what isn't interesting for them.
Their only interest in Blockchain is to track citizens funds movements in real time, so they make sure to raise the income through taxes for the public vaults.
It's the perfect system for a surveillance state. Governments will try to justify their transition towards a full-fledged digital cash system (CBDCs) just to get that they want. In essence, governments and central banks will gain more power. The new permissioned (private) Blockchain system will begin with payments, all the way to proof of ownership, driver's licenses, and more.
I think it's still too early for something like this to materialize. Perhaps, it will take 1-2 more decades before it's ready for primetime. At least we'll have Bitcoin to save us. As long as it remains decentralized, nothing should be able to stop it. I envision a future where both public and private blockchain networks will co-exist. Like it or not, digitalization will radically transform our society in ways that were never thought possible.
