If you add up the port capacities of Newhaven, Ryde, Dover and Folkestone and add some air-dropped stores plus the beach capacities, then you do not have a supply capacity problem. If Deal is taken, that adds a sheltered beach that was once Britain's busiest port. Beaches can be used because there was bad weather on only five days between September 19 and October 20. The piers at Brighton and Hastings can be repaired easily as they only had one span blown up. Beaches were the major source of supplies for Overlord until they took Antwerp. The troops in the first wave would have landed with five days' supplies anyway. the beach capacity was several times that needed by the troops so extra stores could be landed on good weather days to make up for the bad weather days. Supplies only had to be carries a short distance once landed, even to London was only 40-50 miles.
The problem with supply capacity comes when the second wave lands, as the capacity needed would then nearly double, so the second wave must capture Southampton/Portsmouth and/or other ports as soon as possible, as by the time it lands it may not be able to use the beaches at all.
The map shown is for the initial Army plan, which was changed to have an invasion area between Brighton and Folkestone only (but not including those towns), though the objectives were the same and if shipping could be found somewhere the other army group might have been deployed.
The first wave was supposed to take the first objective line and then hold it for up to 10 days while the third echelon is landed along with the air-landed division. The second wave would then start arriving. It took three days for the British counter-attack to arrive at the beaches without opposition in a 1941 exercise but that still leaves up to a week when the troops have to fight without significant reinforcements against a steadily reinforced opponent. The first wave had about 350 armoured vehicles and air superiority but whether it would have been possible to hold out for that long nobody knows.
The invasion fleet consisted of 3-4,000 vessels, not just a few destroyers. The escorts alone were not expected to stop the RN. There was a layered defence consisting firstly of 40 U-boats plus aircraft , then minefields plus aircraft (plus coastal guns at the eastern end), then escorts plus aircraft. They did not have to fight the entire RN, just the Nore and Portsmouth commands, which was quite a considerable force i.e. one old battleship, 50 destroyers (mostly first world war types) and light cruisers, and hundreds of smaller ships and boats.
The campaign may have been a short one. How many government members had Churchill's backbone and inspirational qualities? How many had previously been appeasers? Churchill had to face two votes of no confidence - just for losing Tobruk and Singapore. Then he was voted out of office before the war was over. What might his political enemies have tried if London was under threat? Churchill loved to be in the frontline and would go up on the top of his building to watch the air raids. He might have been killed by bombs or fighting in the front line - as in this story...
“Later that afternoon with the Germans already in Trafalgar Square and advancing down Whitehall to take their position in the rear, the enemy unit advancing across St. James 'Park made their final charge. Several of those in the Downing Street position were already dead... and at last the Bren ceased its chatter, its last magazine emptied.
Churchill reluctantly abandoned the machine-gun, drew his pistol and with great satisfaction, for it was a notoriously inaccurate weapon, shot dead the first German to reach the foot of the steps. As two more rushed forward, covered by a third in the distance, Winston Churchill moved out of the shelter of the sandbags, as if personally to bar the way up Downing Street. A German NCO, running up to find the cause of the unexpected hold-up, recognised him and shouted to the soldiers not to shoot, but he was too late. A burst of bullets from a machine-carbine caught the Prime Minister in the chest. He died instantly, his back to Downing Street, his face toward the enemy, his pistol still in his hand”
In any case, IOT the invasion was impossible because it could take place no later than late September, which means the Battle of Britain had to be won by the first week in September, which means the main part of the Battle of Britain would have had to have started about a month earlier... so Germany had already lost the war when the BoB started.