Hitler's rash decision to change tactics saved the nation.
Having studied this from a lot of angles, I'm not so sure it was Hitlers' fault per se. (Does that invoke Godwins' law already?)
During the battle of Britain the Luftwaffe were systematically destroying our early warning radar, and bombing airfields, pushing the RAF to more northerly air bases. Over time, we'd have had no answer for this, and been unable to prevent any invasion force, or keep control over the channel via the Royal Navy, as the Navy would just have been bombed to bits.
Due to an accidental bombing of London by a lost german bomber, Churchill had an excuse to retaliate and bomb Berlin. Very minor damage, but Goering/Hitler had promised that Berlin wouldn't be bombed.
That left them with 2 choices - lose face, and keep destroying the RAF, or bomb London. Nazi pride being what it was, the focus switched to bombing London, and allowed us to sort out airfields and train up new fighter pilots in time.
A very wise decision from Churchill, now that we have the benefit of hindsight. No detriment to the brave who fought in the skies, without those, no change in tactics would have helped.
They will be remembered.