The Luftwaffe had about 2600 planes in 1940 and the ability to build as many airstrips as they pleased all along one border of the Channel. What do you think the chances were of a British ship or task force's surviving a single mission into the Channel? I agree with what you're saying that the Brits would've been willing to risk their entire Navy to stop an invasion of the homeland (which BTW was the original reason for the creation of their Navy centuries ago) but would that have been enough? (And if they lost most of their Navy, would they have been able to continue to feed their population? Remember that the Brits were dependent on food from overseas.)
If some power wants to attack a US aircraft carrier, they face the formidable task of finding it first. On the open ocean this is very hard: during the Cold War, the Soviets deployed an entire constellation of satellites for the sole purpose of locating US carriers in the North Atlantic (then transmitting the location to their submarines and long-range bombers). I mention this because the reputation for awesomeness earned by the British Navy before WW II was earned mainly in unconfined waters; against air power and subs, it is much less powerful in the confined waters of the Channel.
Also, look what happened to the British Pacific Fleet the next year when it dared to sail within range of Japanese aviation assets. And I think that happened in unconfined waters.
Honestly, I figured the GP would get a kick out of reading about a hypothetical war game of the scenario they envisioned.
Um, yes — I think I mentioned that.
Naval ships also are just a bit useful for control of the sea; indeed, historically that's been arguably the foremost purpose of the navies of island nations such as the UK and U.S.
If the Germans had tried to invade Britain, the Royal Navy would have thrown in every last ship it had to attack the German troop transports in the Channel. (I'm paraphrasing Herman Wouk here.) The RN ships would have been extremely vulnerable to U-boat and E-boat attack there, of course (just as the U.S. Navy and RN are worried about in today's Strait of Hormuz). The RN would also have been vulnerable to air attack (as the British, U.S., Dutch, and Japanese navies learned at terrible cost in the Pacific War), especially if the Luftwaffe had previously gained mastery of the skies over the Channel. But as the saying goes, desperate times call for desperate measures.