Because there is no reason to do so? Providing e-mail address is service successfully provided by private sector, no need to use public money to destroy competition in private sector and centralise a service to one provider.
Most of EU engagement in digital services is to encourage market sector, not replace it with a public monopoly (see e.g. eIDAS).
Most of EU engagement in digital services is to encourage market sector, not replace it with a public monopoly (see e.g. eIDAS).