I have to wonder if the Japanese Military felt that the Zero needed to be replaced?
It appears a viable question, but, yes, the IJN had definitely intended on replacing the A6M, with the A7M Reppu, which should have had a similar timeline as the F6F, but for many seemingly inexplicable reasons it didn't. It's first flight in 1944 was not until after the F6F had already entered service aboard US carriers, two years after the project to replace the A6M was reinitiated in mid 1942 - it had its origins in a requirement laid down in 1940 around the same time as the A6M first entered service, and why was this not progressed with in the mean time to be reinitiated two years later?
I have not read any definitive answer to exactly why the resources were placed into improving the A6M instead of concentrating on getting its replacement into service. From the reading of English texts, which might suffer owing to lack of credible information, the tactical situation the IJN found itself dealing with following the defeat at Midway and the loss of a significant portion of its carrier fleet saw a rise in urgency for land based interceptors and effort went into those, Mitsubishi concentrating on the J2M, whose development began before Japan went to war, and Kawanishi the N1K land based fighters, while the A6M was left to soldier on on carrier decks.
Development of the A7M was protracted, with bombing and a natural disaster contributing to its failure to see production before the end of the war, but these things did not begin to affect the programme until 1944/1945, so what happened beforehand? Was it as simple as a lack of carriers meant that Japan's focus changed to land based interceptors keeping the enemy away from its newly captured territory? It seems a stretch to believe such a simplistic view of a reaction to the strategic situation Japan found itself in, but Mitsubishi's resources did focus on the J2M and improving the A6M, rather than concentrating on pushing the A7M toward service status. One author hints at the fact that Mitsubishi was overworked and under resourced but that doesn't explain the effort stream channeled toward improving the A6M instead of developing the A7M...