Not sure what you mean by FAA 'sorting out its defects'. The issues with tail wheels failing and 'assymetric stall' were pretty well sorted out as the FAA squadron 1830 was being formed in San Diego June 1943. VF-17 did its shakedown cruise on Bunker Hill in July. That said, VF(N)-101 were first F4U squadron to fly combat (Enterprise) in January 1944 - but it was a night fighter unit. About the same time FAA went operational in Med. The F4U was finally 'certified' for US Carrier ops after successful mod to landing gear in April 1944.
AFAIK, the claim re: FAA sorting out the F4U was primarily based on the claim (false) that FAA introduced 'new approach' flight pattern to provide better visibility until final flare and landing but those processes were introduced in early 1943 before FAA even received 1st Corsair? The primary issues with precluding F4U early in carrier qual process were 1.) upwash inboard wing stalling before downwash wing, 2.) tail wheels blowing out, 3.) severe bounce due to original oleo strut design, d.) loss of rudder authority at low speed. All design, not 'process' issues. All sorted out by Vought, not FAA.
What is absolutely true is that FAA was operational earlier and in greater numbers than USN carrier ops. Further FAA was deploying all of its F4Us from Carriers when USMC/USN operated most F4Us from land until fall 1944.
I think it's one of those stories which gets a bit exaggerated, but there is some truth in it to the extent that the institutional inertia created by the early problems with the F4U in Navy / carrier service (some USN units flew them from land) was put paid by the obvious success FAA was having with them flying from carriers, with the benefit of the improvements Vought made. The FAA worked out their own approach and as we know, also had slightly different Corsair variants with the clipped wings etc.