The current option of going "paperless" by way of providing a privately owned email address is a huge risk factor. Gmail should not have access to view my government related emails. They aren't accountable for network issues either that may result in personal damages. The email itself is commingled with spam streams, related filters, and other things.
We need a dedicated, government protected, digital communication line. Sending mail through it would be tied to your government identity. Behavior is then open to regulation and accountability. Which is a good thing for the specific need of two known identities talking to each other.
Google can be the contract vendor for the service, that's fine. But there needs to be a contract setup where they are simply providing the service and not owning the data / contents. Maybe Microsoft outbids them the following year, and the underlying provider changes to no observable effect to the users. that kind of thing.
As a bonus, it would be immensely helpful for people dealing with permanent housing issues and the host of problems that are associated with requiring a mailing address for things. A government regulated email address would be sufficient for a lot of those things.