The worry was that the Germans would have been able to isolate both bridgeheads - Utah and the British/Canadian - and destroy them separately. The 12th SS Panzer tried to separate the Canadians from the British on June 7, so the Germans might be expected to try the same thing with the bigger opportunity of No Omaha. The Allies made every effort, post 6/6, to link the bridgeheads together. And in hindsight, having Omaha available, after the June 19-21 storm destroyed the Mulberries, was crucial to supplying all of the forces.
No one can know, of course. Naval gunfire saved the American beachheads in the Sicily, Salerno and Anzio landings, and it might have done the same in Normandy. But it would have been ugly.
- oldman