The Soviets stopped catching up to the west under Brezhnev, and never recovered from what Gorbachev called, "the era of stagnation."

The Cold War was just one of many factors leading to the Soviet Union collapse. It was a lot more complex than “stopped catching up to the West”.

Source: I lived through it.

Perhaps it would be more accurate to say it was the biggest external influence on the collapse of the Soviet Union.