To be fair, the F4U-5 was using a fully variable speed supercharging system: Pratt and Whittney themselves categorize the -8 and 8W on the -1 Corsairs as a "B series" Double Wasp, the -18W and -42W as "C series", and the -32W of the -5 Corsair as "E series". A LOT of extra work had gone into refining the power output of this engine, and in this case, it meant that much of the extra power came from better supercharging
Another thing is that the extra "juice" on the P-51H compared to the wartime D models was brought on by water injection, boosting maximum allowable manifold pressure to a staggering 90". To give you some reference, the P-51D with 150 Octane fuel was cleared for no more than 75" in USAAF doctrine, and the RAF cleared V-1 chasers up to 81". Water injection doesn't so much add power to the engine as it does allow the engine to run closer to full power without risking detonation. The Corsairs from the -1D onwards also all have water injection.
A supercharger ADDS boost and brings you closer to the limit of the engine itself. Water injection increases the limit of the engine, thus increasing the maximum safe boost
Therefore, at higher altitudes, the the maximum boost that the second supercharger gear can give eventually drops below 75", and at that point, the power is essentially the same as a P-51D. Yes, the Corsair also uses Water Injection for these figures, but it's particularly noticeable on the P-51H due to the sharp increase in power, combined with the Corsair having had Water injection for more of its life, so the characteristics of water injection are more "baked in" to what we think of in the Corsair's performance
As far as climb rate goes, I'm somewhat confused by that myself. As far as I can tell, the -30W engine shouldn't be THAT far behind the -34W at low altitudes to explain it falling behind the -1 to such a degree. For the Mustang itself, they shaved down the weight a LOT by using what they learned from the P-51D. In particular, the Landing gear system alone was around 300lbs lighter by switching to aluminum parts rather than the steel used on the D. Of course, this had the unpleasant tradeoff of making it far less suitable for rough field operation, and the tail wheel was notoriously buggy, to the point that later on, many P-51H's simply had the tail wheel locked open. NAA did a damned good job, but even though a Naval variant was considered, with the landing gear issues, I think they made the right call by sticking with the Bearcats and Corsairs for their needs