"...not broad enough field of fire?"
When you say that, it sounds like you're assuming a gunner would be there to traverse the gun, which would of course have been impossible in a Spitfire or a Hurricane. And a true unmanned tailgun would assumedly be aimed down the centerline of the airplane, and its "field" of fire would be zero. It would be able to shoot only at an attacker literally and exactly in the airplane's six o'clock, and it would have taken the Luftwaffe maybe three minutes to learn that one never attacks a fixed-tailgun-equipped airplane from absolutely directly behind.
If you're imagining a manned "tailgun," it would for weight-and-balance reasons have to be from a position aft of the pilot, and that was tried. They were was called the Fairey Battle and the Boulton-Paul Defiant and were dreadful failures.