Good idea which will work for most people, however it'll be an issue for dissidents or people moving countries. Those can use some other provider, while everyone else (99%?) can be covered by this solution.
I guess there should be a provision against gov services making it mandatory for people to use the gov provided email.
A government that would punish dissidents by locking them out of bank accounts etc. can already do so by extrajudicially freezing their bank accounts in the first place (looking at you, Canada).
In my view those email address should be strictly national and reserved for national-level services, so whenever you move to another country you should get a new email address at the same time as you get a tax ID number.
I guess there should be a provision against gov services making it mandatory for people to use the gov provided email.