The securing of the beaches also meant the denial of causeways leading to the beaches. An invasion on a broad front had to be accomplished fully to prevent traffic jams (which occurred anyways as resistance inland was fierce) and to deny the Germans a salient from which to hit the flanks of a beach-head. The deliberate avoidance of assaulting a port directly meant that as much real estate as possible would be needed; especially since the Allies had to begin landing supplies directly onto the beaches in addition to their improvised harbors.

Consider: If say, Omaha beach repelled the assault battalions back into the ocean (as indeed, Bradley at one point feared would happen), the units destined to land at Omaha would instead jam up into Utah. The Utah beachhead would then have been open to attack from its flank from the still-secured Omaha causeways. You have to understand the logistical difficulties of defending a beachhead on two fronts while simultaneously attempting to expand it.

As for pockets of Germans surrendering...unlikely. For reasons of politics and (arguably) sound military principles, pockets would hold out for as long as possible; and evidence for this on all fronts are manifold. Leaving large pockets of Germans in such a critical area of supply would be unimaginable to a military planner, and considering the beaches necessitated a steady build up of manpower, would be impossible to crush in an acceptable amount of time. It would be a dangerous tying down of forces that would need to be pushing in land and setting up blocking positions for inevitable counterattacks.