They kept them for their high altitude interceptor units.
That is what they acquired them for in the first place. For close support you need a slow expendable plane like the IL2. They built thousands of those, so why waste your few best high alt fighters?
There are two things that made the P-51 such a success: one is the large fuel tank connected to a relatively fuel-efficient engine, and the other is a modest price tag. True the rest of the package was a decent fighter, but it was generally not better than a late model Spit or P-47 as a pure fighter at high altitudes.
To be a success and get the kills, one needs to reach the enemy. P-47s could barely, even though the one fighter group that kept their Jugs is the most successful one in the 9th air force. F6F ruled the pacific not because it was a better fighter than the F4U - it did it by being there when the big battles happened, while the F4U was still not qualified for deck operations. By the time the F4U did and was slowly rolling out the production line, Grumman has already broke every production rate record in order to fill the navy with F6Fs. At their peak, Grumman produced over 300 F6Fs a month - that is enough to fully equip a new squadron every two days...
Now, this thread is about the best "heavy fighter". I always thought that term referred to heavily armed twin engine fighters meant to destroy bombers while still able to dogfight. For example: 110, Whirlwind, Beaufighter, P-38, Mosquito. It seems that most people here thinks it means the best JABO. In any case, the P-51 is neither. Sending the P-51 to do ground attack is a folly, just as it was to send the Spitfires to do this mission. They simply ran out of planes to fight in the air, while there was still a fight to be won on the ground. Many good pilots lots their lives because of that, but perhaps they saved lives on the ground, so in the grand cynical scheme it was worth it. "The best" tool in a war sometimes just means "available", "cheapest", or "expendable".