I think government email services would be very risky, both in a legal sense and in a cybersecurity sense.
You can have government authentication without too many issues, though. The EU has a digital login system that's federated (EIDAS) and your government should never have to retract your ID unless you die. Some implementations use physical smart cards to sign in, others use usernames/passwords+apps/2fa. The system is currently intended only for official government business, but under the hood most systems use something close to OAuth2 so it should be possible to modify the system to allow generic logins.
The downside of such a system is that the using the same authentication for government grants and your social media makes it very easy to phish people. This increases the security requirements of the system significantly, which in turn make the system very annoying to use. The authentication mechanism also allows the government to track exactly what services you use and when.
Personally, I'd much rather see OpenID return. The ability to pick an identity provider and log in would allow for multiple accounts and the ability to control your own data.
Alternatively, wide-spread use of something like https://irma.app/ would be even better. This allows for decentralised identity that you can take with you, just like a regular ID. It also allows for exposing only scoped data (ie "is this person of drinking age" rather than "what's this person's birthdate").