Would you want to use an email account provided by your government?
State broadcasters are a good point, but we tend to claim they aren't really "by the state" but "by the public", and thus independent from the state. Countries that don't do that (e.g. Russia, China) have invested heavily to have their local versions of Google, Facebook & friends.
Estonia gives a government email address to its citizens. I use it for some real world services, especially related to government. I kind of regret it, because the address is now inundated with spam. Apparently some of these services made that information publicly available.
State broadcasters are a good point, but we tend to claim they aren't really "by the state" but "by the public", and thus independent from the state. Countries that don't do that (e.g. Russia, China) have invested heavily to have their local versions of Google, Facebook & friends.