Yet at some point in the story Ozymandias says that even with the omniscient nuclear guy, the Soviets build so much nuclear weapons - 50 000 of them - that even Dr Manhattan couldn't stop them all. So in a sense, the existence of DR Manhattan makes nuclear apocalypse more probable... for the exact reasons you mention in your post.
The fact is if a government starts spending a lot of money on something in an arms race, the opposing government will have to assume that it is something worth that kind of money. And start devising countermeasures.
Practically that means the Soviets see US spending billions of bucks on a system that purportedly can negate the Soviet second strike capability. They then also see the US calling for across the board reduction of strategic weapons. While having extremely accurate missiles. After connecting the dots, they will conclude only one thing... And the Soviets will not like it.