What are you talking about? Coral Sea, Midway, Philippine Sea, Cape Engano, and Ten-Go were all resolved exclusively by carrier aircraft.
Midway and Coral basically amounted to exclusively a contest between carrier vs carrier, not aircraft vs large, heavy SAG trying to break through to a third target. The Phillipine Sea was largely resolved between the American and Japanese heavy surface forces at Surigao Strait on the one hand and the light surface forces of Taffy-3 vs the heavy surface forces of Center Force at Samar on the other, with carrier aircraft playing a secondary role, which is a FAR cry from your claim of being settled "exclusively by carrier aircraft". Notably the one action within the Battle of the Philippine Sea which does match the description of aircraft vs a large heavy SAG, Surigao Strait trying to breakthrough to a third target, saw the American carrier aircraft
fail to prevent Center Force from penetrating the San Bernadino Strait and had the taffy's not had the semi-happy luck of being in the way at Samar, they would have reached and butchered the American transport ships. The Ten-Go force was a grand total of six vessels, hardly a large force, and one of them still managed to get away. Cape Engano saw 8 vessels sunk of which 5 were by aircraft and took the course of the entire day in which the Americans were in no way pressured to sink these forces before they reached a extremely vulnerable third force, not to mention it was against a IJN force almost 1/3rd the size of the Royal Navy force in the Sandhurst game.
If the Luftwaffe manages to match the American's feat at Cape Engano and sink 5 vessels in a force the size of the one dispatched in the Sandhurst Game in a single day, which is a worst showing then what I posited them achieving
as a best case in my earlier post on this thread, then by the end of the day there would still be 69 Royal Navy cruisers and destroyers who merrily sail into the German barge fleet as it's partially across the channel and murder the ever loving fuck out of it. The Luftwaffe would have thus failed in
stopping the Royal Navy from destroying the invasion force and causing the entire invasion to fail as assuredly as Halsey failed to prevent Kurita from passing through San Bernadino.
The problem isn’t that the LW can’t sink British ships. If the entire Royal Navy were dumb enough to just anchor itself in the Channel and sit there for days and weeks on end, then the Luftwaffe would eventually destroy it. But for the task the Luftwaffe is actually given, the Luftwaffe has to utterly destroy virtually the
entire home fleet
in the short period between which it comes into range and when it would be amongst the invasion fleet. No Air Force in 1939-1945 could have achieved this. The fact the Luftwaffe is also being asked too provide CAS to the landing forces, hold off the RAF, and bomb London all at the same time their expected to perform this impossible task is just icing on the cake.