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Would having additional Battleships, particularly those with larger guns (i.e, 16-inch guns), have made a difference?
It could, but it would have required a very different fire plan.
First note when the naval fires started. At the start of nautical twilight or in laymans terms when it was light enough to see the largest landmarks on the shore line. Roughly 05:15, I'd have to check the books to pin down the exact moments for each beach. There are a number of reasons reasons for this, but the effect was a very short naval bombardment. Barely 30 minutes in the case of UTAH Beach, about 45 minutes for OMAHA beach, and longer increments for each sucessive Canadian and British beach. Second note the targets. The BB were predominatly aimed at the defending heavy batteries, the big guns of the coastal artillery. The cruisers and destroyers were aimed at the lighter beach defenses, the MG and AT cannon bunkers.
Second: Note that during the pre H Hour preparatory fires very little was aimed inland at the German field artillery & other positions. Significant NGF attacks inland did not start until about fifteen minutes before H Hour for each individual beach.
Third: All the German defenders were in their battle positions before 05:00, before the naval and air preparatory attacks started. Most of the air attacks tapered off at 01:00 before the airbourne drops started at 01:30. So between 01:00 & 05:30/06:00 there was very little fire falling on the inland German positions.
Why is this important? The German soldiers defending the coast did not sleep there. They were billeted inland, usually 1 to 5 kilometers. On a ordinary night the MG and gun bunkers were manned by two guards, the OP & CP by maybe two or three watch standers, and a scatter of sentries in between. It is correct a 'Invasion Imminent' warning had been sent out on 3rd June, but that warning had been issues before and given the weather was difficult to take seriously. The soldiers habeen working extended hours at defense construction and training. the commanders were reluctant to damage those two tasks by further exhausting them with a full alert. Some additional sentries were posted & the result was the beach defenses were manned at less than 25% strength at 01:00.
When the paratrooper drops started at 01:30 the alarm imeadiatly went out & by 02:00 orders were arriving at the barracks & sleeping quarters to battle stations. So some 50,000 German soldiers were on the road on foot, horse, and unarmored autos headed to their combat posts. By most German accounts they had all reached their posts between 04:30 & 05:00. Many survivors remembered breakfast was being distributed to the battle positions shortly before dawn.
So, lets assume the NGF ships, including any extras are not quiet until first light. Lets imagine for a moment a fire plan was written to have then fire on the beach defenses, and the approaches from inland from sometime between 01:30 & 02:30. That would then require the German garrison run a guantlet of incoming fires for three hours to reach their battle stations, and do this in the dark.
I'll leave folks to draw their own conclusions on what this might to the strength and morale of the beach defenses when the first assault wave grounded.