>I think the Cold War played a role too.

Yes, definitely. The USA wanted as many allies as it could get against the USSR, even former enemies.

>German factories were, on average, equipped with more modern machines than British ones.

Thankfully the Germans wasted amounts of resources on not very useful weapons, such as the V2 rocket (technologically brilliant, strategically pretty useless) and the King Tiger (unreliable and IIRC cost somewhere around 20 times as much to make as a typical allied tank).