There are several problems with this idea. As you say this infrastructure is important, but it's too important to centralise it and put into government's hands. If there is a government monopoly you will not be able to shop around, choose best service and will have to deal with

* poor service, since there will be no competition and little to no incentives to improve it

* lack of privacy, since law enforcements and random officials will absolutely want to have full access (and sure, Google has access to my emails, but I can use Protonmail instead and trust them way more not to abuse this access than I would trust "DMV for email")

* even further centralisation of power - it would be much easier to apply pressure to people doing anything "anti-government" (I believe there are already examples of this with people being cut from all-encompassing WeChat in China, but I don't have references at hand)

There is also an argument against frivolous expansion of human rights with anything not absolutely necessary or too specific.