In the same way that "Red Arrows" is just a nickname for the properly-styled Royal Air Force Aerobatic Team, so "Battle of Britain Memorial Flight" is also a misnomer; its actual title is just "Royal Air Force Memorial Flight" (check out the Unit's emblem).
I'd agree with most of the sentiments expressed here regarding the apparent relevance of the Red Arrows. The problem with our collective viewpoint however is that, as most of the posters here comprise Roger Bacon's Total Aviation Persons, we are probably the least representative panel to discern its perception by the person on the Clapham omnibus.
The Reds do at least serve a useful purpose for all attendees at air shows; as a finale, they ensure that Joe Public is obliged to sit through the boring Chinook, Typhoon and F-35 displays in order to enjoy the pretty smoke at the end. And whilst they stand transfixed, the rest of us can make a sharp dash for the exits.