I agree the Americans outperformed the British in Normandy. I suspect the Brits had had enough of war by that time and were just looking to get home alive.
But the German army was in a class by itself throughout the war. Time and time again German troops performed better than one would have expected given the relative forces. They chased the British around North Africa for a couple of years with only a few German divisions even though Africa was the main British war effort and it was a minor sideshow for the Germans compared to the Russian front.
Even when American troops were involved such as at Salerno, Anzio, Monte Cassino, Normandy, Brest, Huertgen Forest, etc the Allies were frustrated by just how difficult it was to defeat the Germans. They pretty much always fought hard even when the situation was hopeless as at Brest.
Probably the best example of the uniform excellence of the German troops during the war was at the Seelow Heights in 1945 when the Germans basically cobbled together a bunch of stragglers and Luftwaffe ground crews and stopped a massive red army thrust cold for a period of time. Zhukov couldn't believe what he was witnessing.
You obviously know a lot about WW II, so I have a hard time seeing how you could disagree with this point. It's not surprising that a modern day Sparta (which is essentially what Nazi Germany's culture was like) would produce fine soldiers.
The Germans outperformed per man or per unit of military capital, did not outperform the western allies overall, did outperform the Russians overall. Mostly because of doctrinal weaknesses in the latter case, and mostly because of smarter tactics and a professional officer corps.
They also had lousy strategic direction, were completely outplayed by the Russians in operational "big chess" terms, threw away their armor reserves recklessly in hopeless counterattacks, nailed their infantry positions to the ground and needlessly lost forces by doing so, etc, etc.
And no, the Nazis weren't better at it because they were more Sparta like or more tyrannical or cared less about human life. They lost. They performed well tactically in exactly the same manner the Germans of 1914 had, and the Germans of 1870, and the Germans of 1866. That's the result of a tradition that runs from Frederick through Gneisenau to Moltke the elder and von Schlieffen. Nazis didn't make it, they executed not a few of the men who actually did, and they thoroughly wrecked it, leaving ashes.