>Big navy warships are extremely vulnerable to air attack. Plenty of examples in WW2.

That's true. But there were also plenty of British submarines, destroyers, gunboats, MTBs and other smaller ships. Also Germany didn't have much in the way of specialized maritime attack aircraft (IIRC because Herman Goering didn't want any competition from the German Navy).

The RN certainly would have taken heavy losses from German aircraft, submarines and naval mines (which were planned to be sown both sides of the invasion corridor). But it's hard to imagine that they could have landed enough troops and supplies against a far superior naval force without proper amphibious landing ships. Imagine trying to get tanks and artillery ashore from a barge.