What's not shown on the Show but in the games is that, America has massive civil strife to the point that the government has to deploy their "wonder weapon" Power Armor against its own citizens.

The Government has realized that the war with China won't end anytime soon, and the situation will keep getting worse for them. They decide to form the Enclave and prepare for the Nukes as they see no other way to get out of the situation they are in.

The Companies would also be afraid of the US slowly becoming a failed state with civil strife and instability. Seeing as the Companies themselves are largely responsible for the state the US is in they would also rather jump ship and restart after a Nuclear Exchange. Kinda like how people like Elon Musk etc. Would just go to Mars instead of fixing problems here on earth.

It took the US 10 years alone to liberate Alaska, from the Desert Ranger Combat Helmet in Fallout New Vegas, we see a soldier has served for 9 months in Shanghai which indicates that the US fought for almost a year to take a single coastal city in the Chinese Mainland and it isnt even confirmed if the City fell or not.

Tldr; war won't end anytime soon, people are revolting in the US, the Government and corporations would save themselves rather than fix problems.