The losses in Italy would be a more logical reason why Normandy was invaded; the Germans could have traded 10:1 losses will the allies all the way until the end (as it was, northern Italy was still held by the Axis at war’s end). Normandy by comparison was a much softer target; it was impossible to defend the vast stretches of the French coast in any numbers. Normandy is a much larger battle in folklore than the numbers involved would indicate.
The US wanted a cross channel invasion and said so continuously. The UK fought the idea halfway up the Italian peninsula. The Normandy battle was supposed to not be a big battle, it was supposed to be as easy as possible — establish the beachhead and keep pushing out in all directions. LSTs landed troops and tanks with or without the artificial harbors, and Operation Double Cross kept most German reinforcements from being sent until it was too late. There was no cakewalk in France, but as it turned out the narrow gauge of the Sherman (which was built on auto assembly lines) was ideally suited for those lanes between the hedgerows and the narrow streets of many a French village. Much better than fighting in Italy, and it served an actual purpose.