Understanding John Thatch's "Big Blue Blanket":
Even with radar late in the war a Carrier likely wouldn't spot an approaching enemy until that intruder reached say 50mi out from the fleet. This Kamikaze intending to destroy your carrier is likely travelling at up to 300mph which means it will cover that 50mi in roughly 10 minutes. A Hellcat or Corsair in the hanger at this point is utterly worthless. That plane couldn't get fueled, armed, up to the flight deck, warmed up, and off the carrier before the Kamikaze arrived.
Even if the Hellcat or Corsair was fully armed, fully fueled, and warmed up on the flight deck, it takes 8-10 minutes for a Corsair to climb to 20,000' depending on the load-out. Since you'd want maximum fuel and ammunition you are probably closer to the 10 minute mark which means you'd get to 20,000 at about the same time the Kamikaze crashed into your ship.
Ok, so what if the Corsair or Hellcat was already flying 20,000' over the carrier when the intruder was identified? Well top speed for a Hellcat or Corsair is around 400mph so if it was immediately given the correct compass heading and altitude, it could head toward the intruder at a closing speed of roughly 700mph. That will get you to the Kamikaze in about four and a half minutes at roughly 25-30 mi from the carrier. In theory you could shoot the intruder down head-on but this is an INCREDIBLY difficult shot. The closing speed is 700 mph.
700*5,280 = 3,696,000 feet per hour
3,696,000/60 = 61,600 feet per minute
61,600/60 = 1,027 feet per second.
The M2 Browning .50 cal machine guns in the Corsairs and Hellcats have an effective range of about 6,000' so you have roughly six seconds from the time the intruder is in range until you crash into them. You can't fire for the full six seconds because you have to pull up or else you'll crash into them and you have zero chance of surviving a 700mph collision at 20,000' so you have AT MOST approximately five seconds of firing time. That would be plenty if you were firing at some stationary target but you aren't, everything is moving. You and they are moving at a combined 1,027 feet per second and remember that this isn't a ground or sea based target that moves in two dimensions, this is an airborne target that moves in three dimensions as do you.
Actually hitting an approaching Kamikaze with a head-on shot in this situation would be miraculous.
Alternatively, you could loop around behind the intruder and approach them from astern but that turn takes time. You have to slow your aircraft down enough to make the turn and you have to line it up right and then once you get behind them you have to catch up to within no more than 6,000' before you can start firing effectively. Firing earlier may sound like a good idea but it isn't. Your .50 cal machine guns fire 450-600 rounds per minute and you only carry 400 rounds per gun so you have less than 60 seconds of firing time. Firing "prayer" shots from from the 24,000' maximum range of the gun accomplishes nothing but to expend your ammunition while you are too far away such that when you do get close you will not have any left.
By the time you get turned around and lined up behind the intruder you are probably no more than 20 mi from the Carrier and that is if everything went right. They are still closing on the Carrier at 300 mph (5 mi/minute) so you have AT MOST four minutes to shoot them down and the last minute or more of that you'll be chasing them into your own ships' AA fire which is just as effective at shooting down Hellcats and Corsairs as it is at shooting down Zeros so you'll be at SUBSTANTIAL risk of getting downed by "friendly fire" if you don't splash the intruder before you get close.
Realistically you only get one or maybe two passes at the intruder. You'll be trained to keep your own throttle to the firewall in order to avoid being shot down by any lurking enemy fighters so slowing down to their 300 mph speed to get more shots simply isn't an option. Your closing speed will be at least 100 mph or about 150' per second so you'll have something like 40 seconds overall but you probably don't want to open fire until you are closer and you have to pull up to avoid crashing into the intruder so this still isn't easy. Then, if you miss, you have to either dive or climb away, loop around, and try again with very limited ammunition.
Corsairs and Hellcats are also armed with a couple of 20mm cannons but those have an even higher rate of fire and less ammunition than the .50cal guns so they generally aren't engaged until you are pretty sure you are on target.
Placing DD's and DE's 50 miles from the Carriers and protective Combat Air Patrol (CAP) fighters at long range as well moved all these calculations an extra 50 miles away from the Carriers which gave the fighters more time to get their shots lined up which saved a lot of lives on the carriers but as
@Cincydawg pointed out, the DD's and DE's sometimes paid a VERY high price for that.