But the Germans had to win the air war to ensure they would be able to sink the Royal Navy when they sailed into the channel to destroy the invasion flotilla. Without air superiority they wouldn't be able to stop the RN from entering the channel and that is one of the reasons they gave up.
I remember reading a book a long time ago based on a wargame of Operation Sealion, Adolf Galland played Goering and other experts played their respective characters, with the actual weather reports for the summer of 1940. From the wargame if the Germans invaded without air superiority, they would have landed in Southern England and done some serious damage, but the RN would arrive in the channel and although being almost totally destroyed they successfully stopped the invasion in its tracks.