I am also German and highly respect the sacrifice of the Allied soldiers that day, some of whom paid the ultimate price to defend their country and to rid Germany and Europe of the scourge of National Socialism.
Notwithstanding, many people don't know that most of the fighting in Europe duing WWII was actually on the Eastern front (Germany vs Soviet Union), proof of that is the 10+ million Soviets killed in combat and that Germany had 5 times as many divisions fighting East as compared to the West - even after Normandy. There were also 5 times as many Germans killed on the Eastern front compared to the Western front (any history guys please correct me if I am wrong, because I am going from information in my head learned in university and from relatives).
So in my opinion its false to suggest the war was won because of Normandy, Germany was losing it on the Eastern front way before June 1944 (basically since Stalingrad in early 1943). But were it not for the US/England's Normandy invasio, all of Europe may have turned from National Socialism totalitarianism to Soviet Communist totalitarianism. The incorporation of Eastern Europe into the Soviet Union after the war would present evidence toward that hypothesis. Had the Russians been the only major power on continental Europe after the war, who knows what current Germany, France, or Italy would like like today. Who knows how the cold war would have turned out with that scenario.
I don't mean to diminish Normandy at all, I just mean to put it in perspective. Its the Russians that really paid most of the price. They suffered the most military and civilian casualties by far.
The invasion of Normandy was an incredible feat, considering the weather and German defenses. The bravery of those soldiers that day, should never be forgotten.
Some of my favorite movies to this day are still 'The Longest Day' 'A Bridge Too Far,' and 'Saving Private Ryan.'