Why? it forced the Nazis to divert significant troops from the Eastern Front, and was an impressive technical and logistical feat on its own.
I agree with you it didn't single-handedly win the war or anything like it's portrayed as in The Dirty Dozen or movies like that, but it's not difficult to image the Eastern Front bogging down into a stalemate without the threat from the Allies in France (and Italy) tying down another hundred plus German divisions.
The Eastern front wouldn't have bogged down into a stalemate in after Spring of 1944. Especially with Hitler constantly interfering with defensive planning which cause the best lines of defence to be often unenforced or near completely abandoned. Hitler constantly refused to lose ground to gain an improved defensive position since his little 'miracle' at Moscow. Even with the front narrowing it wouldn't have been sufficient.
The last German ability for an offensive in the East was Kursk in Summer of 1943. That was the last gamble, they would never reach sufficient force to mount an offensive, let bring the divisions on the front to full strength.
One of the major drawbacks for the German army was the heavy reliance on horses for supplies. Deep area penetration allowed a small number of heavily mobile units overrun more static units at the start of the war in Poland, France, Yugoslavia, and the initial Soviet campaigns. With exception of the British who were mechanized, all their opponents were heavily reliant on horsepower. The advantage was how the would utilize their limited mechanized units to full advantage. Even at the start of the Soviet campaign the Germans lacked trucks to transport supplies. That includes the thousands taken from France both civilian and military. However, when they broke down there was not sufficient spare parts nor production to recoup the losses in the east.
One decisive turn on the Eastern front which is overlooked is the mechanization of the Soviet forces, in particularly, supply and troop movement. This mechanization was led by trucks via the land-lease program. As the war continued Soviet mechanization increase, the German mechanization decreased.
The fields of eastern europe favor mobility, hence, the cossak/cavalry tradition in the region.
Deep are penetration is really quite simple. Attack along a broad front. The moment a section breaks you pull mechanized units in force to advance as fast as possible by passing as many enemy positions instead of fighting. Meaning the enemy would be forced to abandon entrenched positions to engage or fear being overrun or encircled. The slower infantry units would than follow up to advance in attempts to reach the spearhead and potentially encircle enemy positions.
What is easy to see studying the Soviet offensives late in the war is the period of consolidation of territory following the initial advance. Early in the war Germany was blessed with having their enemies capitulate before true consolidation could take place. First true examples can be seen in the initial Soviet campaign. You need to consolidate land to clear it of enemy combatants to keep a clear supply line, solidify a front line, and start stockpiling supplies for the next offensive. Remember that supply throughput is greatly increased with mechanization. The Soviet offensives of the late war period are easiest to study and see this. If a solid period of consolidation is not accomplished you are vulnerable to counterattack due to an inadequately staffed front line and insufficient supply close to the front.
Not to mention the majority of German divisions on the Eastern front were below 50% combat strength before the Soviet summer offensive.
The Eastern front was mostly plains intertwined with a few river. There are no mountains until you reach the Caprathians along the Polish/Rumanian frontier. The mountains were bypassed with a split between the Ukrainian and Belarussian fronts. 1 going north above the other going south. The front could have held out considerably longer had Hitler allowed is generals to pull back troops to more easily defendable positions. That being said, the pattern from late 1941 onwards said this wouldn't have happened.
The Western allies knew the war was already won by the time the invasion was launched. What wasn't for certain is if the actual landings would be successful. The longterm goal was to block the Soviet advance for their armies were not going to halt in Berlin.
As you said, yes the POTENTIAL of invasion kept a very significant bulk of German troops back in the West/South. However, the invasion did not cause a signifiant drop in forces on the Eastern front. The potential was of invasion was more important than it actually occurring. The actual landings were more important in trying to check a Soviet advance than the war effort itself.
Realistically there was no threat from Italian front other than air bases. There is a reason Northern Italy was still in German hands in May of 1945. Mountainous warfare favours the defender. Even had all of Italy fallen, there are no easy ways out. Into France you must cross the mountains which the Italians couldnt do after joining the war late seeing France was already defeated. Directly north you have Switzerland. North-East you still have the Alps on the Austrian side making any advance slow and nearly impossible for a large force. You could try directly East into Yugoslavia which again has mountains. Tactical retreat in Italy was a way to minimize air bases in Italy as a land force advancing from Italy was no threat to the Reich. Air bases in Italy increased the failed stategic bombing policy of the Western Allies.
The failure of strategic bombing can be seen in German tank production. German tank production actually began to skyrocket in the final two years of the war when Speer became minister of armaments. Had strategic bombing actually been effective as first theorized in WWI, production would grind to a halt and effects would have been seen on the battlefield. However, they were more successful in killing civilians and making life miserable for civilians with very minimal military impact.
A German division on paper in 1944/1945 was a farcry from what one would have looked like and fighting potential compared to 1939. Many 'divisions' existed only on paper having their fighting strength almost completely depleted due to lack of reinforcements. That is primarily true after any Soviet summer offensive 1943 onward. Additionally, many of the nearly created divisions were never even close to 25% combat strength when deemed that. Perfect examples would be the last SS divisions being made. And further, some divisions created late in the war had very questionable combat abilities such as the Luftwaffe field divisions, 'static' divisions etc. The luftwaffe field divisions for example were simply excess Luftwaffe ground personnel thrown into a combat unit with little training nor effective combat leadership.