No, that would lead to even worse abuses.
Cf the present day Hong Kong protests, and Yellow Vests protests - in both cases, the government would use snooping power to spy on the communications, dragnet style. Anybody seriously suspected of partaking would have the account shut down. Since the account would be "the official one", that would most probably render them unable to access other major services, like banking, judiciary, or healthcare records.
Having a corporate steward of email is one more level of indirection. While imperfect, the corporations do put up at least a bit of resistance against governmental over-reach. Even just the fact of having to create, send & monitor official correspondence to demand snooping & take-downs is slowing down the snooping a bit.
Moreover, it would be highly probable the government would mandate a "real name" policy, or otherwise tie the email service to your offline identity. Cue the abusers by online platforms like Facebook & other advertisers.
Lastly, it's more feasible to compete with corporate services than with tax-paid services. And it's competition that drives innovation, quality increase, and cost reduction.