No. It went deeper than that. The British knew that the order of the invasion beaches would dictate the order in which the armies advanced side by side. They chose the leftmost beaches because they would be closest to the coast as they all moved into Germany together.
Churchill wanted to be able to pull the British soldiers out if there was a debacle. Dunkirk was still fresh in his mind.
Besides, the Americans had more vehicles in their divisions. They could travel further. And they did.