Not exactly. It was much less about the US, British and French occupied zones (I think Belgium had one too?) and much more about the rift between East and West Germany.

For the most part it isn't really inaccurate to say the US ran the show in West Germany, especially so far as the de-Nazi-ification was concerned. France and Britain were more occupied with their own affairs.

Whatever regional identities you found in these regions in West Germany were something that existed before occupation. What they suffered, they suffered collectively. The French weren't particularly worse than the British, weren't particularly worse than the US, and so on. Didn't matter where you lived because everyone was being rationed as little as 1000 calories of food daily, didn't matter where you lived, winter was cold, and you'd trade your family's heirlooms if it meant staying warm (assuming someone hadn't already looted it.)