Currently those are provided by private corporations. But will, or even should, governments step in and offer this service to their citizens just like with regular mail?
Came to this idea thinking about the development of the postal service and immediately thought of it as a natural continuation/evolution, but the more I think the more gotchas I see with it. Like what to use as the email identifier (people change names and places, random number are hard to remember, etc), how to stop bad actors from spamming everyone, what if someone uses it to send objectionable content, etc.
The current state where they tell you "to use the internet connection of a friend" when you don't have internet is kind of ridiculous, especially for migrating people from foreign countries that are overwhelmed by European bureaucracies.
You should try getting an internet contract when you don't already own a smartphone. It's impossible, because all ISPs require some form of web identification with shitty apps and your passport that's being recorded.
In Germany we had the "de-mail" for a while with exactly that purpose. But in a government fashion they ridiculously fucked it up, using a proprietary encryption and signing mechanism that never worked up until the point that de-mails were blocked everywhere and all "licensed" providers of that stopped servicing it.
The US government, in my mind and in practice, appears to need 3rd party providers so they can subvert the 4th Amendment under the auspices of "free market activity" in such they just buy peoples activity data where they would have once needed a warrant, or would need one on a government provided platform. Plus, if you aren't a US citizen/greencard holder, you shouldn't be labelled as such as a means of reducing propaganda and conflating worldviews with American views.
Government "spying" is a hard but not unsolvable problem, rooted in our own values. We're so culturally punch-drunk we've no hope on making progress right now. Maybe when the wind changes again...
I don't mind private corp. email, as long as it doesn't capture anything. Basic communication seems such a no-brainer that speculating to make it available to the market seems superfluous.
It's not interoperable with the regular email system (can't communicate with GMail addresses etc), and it's not fully run by the government but by accredited partners. It's meant for sending legal documents across De-Mail users like government agencies and businesses.
But it's barely used.
In Denmark there is a system called Digital Post for communication with public authorities, mostly used for receiving relevant notifications/updates/documents.