I find that difficult to believe.
The F6F operated mostly from CVs and only during the final two years of the war. How did F6Fs manage to shoot down over 5,000 enemy aircraft during 1943 to 1945? Total Japanese aircraft production was about 40,000 during 1943 to 1945. However most Japanese aircraft operated in places like China, Manchuria, Indochina, East Indies etc. where USN CV aircraft weren't likely to be encountered.
Do you have some other operational results to report? And be assured, the F6F did not only operate from CVs. There were land based F6F squadrons in combat, both USN and USMC, not to mention those operating in CVLGs and CVEGs.
The number 5,168 is somewhat of a moving target, I'd admit. US Naval Aviation combat aviation statistics compiled in 1947 or so mark the F6F total credits as 5163 fighters and bomber types only. If you look a little deeper into the report you can also find just for the period 1 Sep 44 to 15 Aug 45 a total of 3,518 credits to F6Fs, this includes 2,278 single engine fighters, 36 land based single engine reconnaissance, 515 bombers/torpedo bombers, 89 floatplanes, 530 twin engine combat types, 17 flying boats, 36 transports, and 17 trainers. Were one to add just the land based single engine reconnaissance, floatplanes, flying boats, transports and trainers from this last year of the war to the gross 5163 reported fighter and bomber credits, the F6F total rises to 5,358, higher than that which you protest and still does not count any of these types credited to F6Fs from their introduction to combat through 31 August 1944. And to forestall the question, these numbers do not include any destroyed on the ground.
Frank Olynyk's compilation of just USN awarded credits comes to 5136, giving a pilot's name, date, place for each credit for an aircraft destroyed in the air as well as probables and damaged. Frank's numbers also do not include any destroyed on the ground. If you add the USMC F6F credits for enemy fighters and bombers from the combat statistics compilation, 93, to Olynyk's USN F6F total you get 5,229, again, higher than that which you protest.
Overall, the Naval Aviation combat statistics reports the aerial destruction of a total of 9,249 enemy aircraft over the course of the war.
If you want to cite Japanese production, then I'd suggest you look at the end of the war. A SCAP inventory of remaining Japanese aircraft reports (combining both army and navy): fighters - 4,554, bombers - 1,790, reconnaissance - 1,076, transports - 169, trainers, 6,681, and other not specified - 2,097 with no report for flying boats, a total of 16,367; this end result from a total production, from 1 January 1941 through 15 August 1945, of 69,270.
Do you have some other sources?