In practice, the .303 MGs proved disappointing in the BoB, which is why the RAF was so desperate to fit cannon even before they were ready. Luftwaffe bombers made it back to base with as many as 300 bullet holes in them.
Probably the worst way to get my point across -- but I think a fair amount of this is 'a poor workman blames his tools'. The flight and gunnery training for RAF fighter pilots was at its absolute nadir at this stage of the war. While reading memoirs / anecdotes of this period you can't help but stumble across this type of thing time and time again. From Alan Deere's book:
Pilot Officer Mick Shand, who came to my flight, certainly looked promising until I learned the grim details of his flying experience ...
"Hello," I said greeting him at dispersal that afternoon. "I hear you're a fellow countryman of mine."
"Too right, and glad to be here," was the cheerful rejoinder.
"What flying experience have you got?"
"I've got a total of 140 hours approximately, mostly on Wapitis. I only managed to get twenty hours on Spitfires at OCU and it was hardly enough to get the feel of flying again after two months' lay off. As a matter of fact I know damned all about fighters, I was trained as a light bomber pilot."
"Have you fired the guns in a Spitfire yet?"
"No, I haven't; apart from a very little free gunnery from a rear cockpit, I've no idea of air firing."
"What about the reflector sight, do you understand it?"
"Haven't the remotest idea what it is, much less understand it."
"Good grief," I thought. "It does seem a shame to throw him into battle so soon." There was no alternative, and certainly no time or aircraft to fit in any further training ... This was typical of the replacement pilots arriving in the squadrons, desperately keen to get cracking but all lacking in basic flight experience.
To me this doesn't strike me as a good environment to judge things. Hurricane night fighters repeatedly proved that eight .303s were perfectly capable of downing a Heinkel with a 1-2 second burst at very close range.
Now I'm not suggesting switch to heavier calibre guns wasn't the correct move to make, but I don't think eight .303s was a bad armament at all in 1940.
According to operational research, the percentage of successful combats rose steadily throughout the war. I don't have figures from the period in question, but in the Spring of 1942 the average closing distance in combat was 570 yards with 12% of attacks being successful. Only 4.5% of attacks were closed to under 200 yards, with 62.4% not even closing to within 600 yards.
By the winter of 1943 the average closing distance was 331 yards and 27% of attacks successful.
By the end of the war this was down to 280 yards and 54% of attacks successful.
Of course there are other reasons for this increase in lethality, but firing range seems to be the major factor in every time period.
Quite a change from 1942 to 1945 considering the Spitfire's armament didn't change all
that much (I left out gyro sight figures from my 1945 totals). I think if the '40 RAF had the luxury of '45 RAF training, tactics, etc. -- the .303 Browning wouldn't have been maligned so much.