The system would have to be built so that the government can’t connect the user to the website, as you don’t want the government to build profiles on website usage by person. Though the bigger challenge here is trust - even a technically perfect system could be circumvented by the operators.
A good example for this were the COVID tracking apps. The approach was built so that as little information was leaked as possible.
What you say makes sense for a multiplayer-only game! But the game has a full single player campaign. There is no technical reason to remove access to that part. That part can be kept working without incurring recurring costs.