Yes - domain registries should be run by the government.
DNS etc can be run by third parties
DNS etc can be run by third parties
On the other hand, if you want to have a say in how other registries are run, then consider joining an at-large structure (ALS) local to you or the Non-Commercial Stakeholders Group: https://www.icann.org/resources/pages/civil-society
Besides, which government do you want to run .com domains