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.

If it were e2e encrypted or had the same rules as the post office has to follow about when they can spy on your mail I would be fine with it.

The post office tracks who sends mail to who for everyone.

I would use one as a source of universal online ID in account recovery, not for everyday use

For official functions yes. As for trust, i trust my gov more than google