Was German rearmament the reason, or something else?
Short answer to a long question, and I'm sure it won't make everyone happy but c'est la vie.
Following WWII, America decided that Paperclipping vital Nazis out of Germany, and using ex-Nazi assets was preferable to Stalin bringing his armored horde through the Fulda Gap.
OSS, later CIA, has zero assets inside Soviet Russia. They coopt Reinhard Gehlen. Gehlen, a general in Wehrmacht intelligence, provided security for Treskow's 'General's Bomb Plot' against Hitler and had assets that had direct knowledge of Soviet Russia. Therefore OSS/CIA cut a deal with him.
Gehlen was given a million and a half US WWII dollars to build the 'Gehlen Organization', comprised of large numbers of SS and SD operatives. They provided Washington with 'details' about Soviet Russia that exacerbated the Russian threat to make Germany indispensable to American plans to keep Western Europe free of Communism. The 'Gehlen Org' will form the core of BND as OSS was the core of CIA.
The Gehlen Org is known to have manipulated data to inflame tensions during the Cold War for the benefit of resurgent Germany.
One of these events is the 'Missile Gap', where Gehlen provides credible information that the Soviets will leapfrog the next wave of bomber aircraft launching nukes and pour a huge amount of resources into creating a missle gap that will defeat America's attempts to hold back the Red Menace. Eisenhower commissions the Gaither Report, which uses Gehlen's facts and figures to plan America's next move.
The US then POURS a ridiculous amount of money and research into building missiles, arms and armaments they will never use and we all pray to God they never will.
When the truth arises, CIA merely points at Gehlen and shrugs. But it takes two to tango.
This leads Eisenhower to make his famous exit speech from the Presidency talking about the danger to freedom from the 'Legislative-Military-Industrial Complex' growing so powerful the tail would soon begin wagging the dog. His political advisors demand he drop the 'Legislative' part of this argument, and the 'Military-Industrial Complex' was born.