I might be wrong (and WILL be corrected if I am, LOL) but I believe one of the main importances of taking Omaha was that it was necessary to create a continuous land presence with the british troops to the east, and the other American troops to the west and link them all up, creating a solid Allied toehold on Europe. Otherwise it would have been easy for the Germans to divide the Allied troops and eliminate them piecemeal. Unfortunately, Omaha Beach was much heavier defended than was thought, and a lot of things went wrong to boot.
Yes, there were enough German reserves that without continuous coverage, the Germans could have split the Allied lines and started flanking the weaker half of the invasion and pushed it back into the sea.
There were considerable German reserves further into France.
What saved us was our ability to command the skies and send in fighters to strafe and attack trains, truck convoys, troop columns at will. Not much is written about the number of fighter sorties launched in support of Normandy, but it was considerable effort on the part of the fighters and medium bombers to eliminate German reserves further inland.