Marshall, the USAAF data I have seen puts the climb rate of the P-47D with paddle blade and water injection at 3,200fpm. (Not bad for a 14,500lb aircraft and slightly better than an F4U-1) At about 12,000 feet, the climb is down to 3,000fpm. At 20,000 feet, she's down to 2,700 fpm.
No, that won't beat a spitfire. Not by a long shot. Perhaps he executed a zoom climb.
At any rate, the P-47's improved climb rate allowed it to stay on an axis fighter just long enough to bring its guns to bear. The paddle blade gave it a 400fpm boost which is not insignificant.
Think 6,000 rounds per minute coverging at 300 yards into a three or four foot circle. It doesn't take more than a fraction of a second to inflict serious damage on your opponent.