Sorry, but you are confusing escort carriers with fleet carriers. Escort carriers were slow-moving, relatively cheap and quickly constructed small carriers mostly used in ASW operations and convoy escort duty. They usually had small number of aircraft, which were equipped with weapons which were effective against subs and that's it. Not that it wasn't important - it was very important - but escort carriers had little to do with fleet carriers. The USA built ~30 non-escort carriers during the war, not "hundreds and hundreds".
Also, you gave the Battle of Leyte Gulf as an example. It can actually be a bit misleading, because this battle actually included several engagements in several different places - in fact, it could be divided into several battles. The Americans didn't use a doomstack or sth - their forces were divided into several fleets. Of course IRL thousands of aircraft could be attacking the same target, but they weren't doing that at once, because that would be pretty pointless - they were attacking in waves.
There are several problems with the game mechanics. CAGs are just naval attachments in HOI2, so in a way they are treated as naval units. However, even if that wasn't the case, HOI games represent air warfare in a very abstract way, so you have stacks of planes which get "stacking penalty" and this leads to strange results, because 1) the damage is spread among several air units and due to the way "hits" work in-game it's much harder to damage a given unit severely, even though IRL a group of air aces could wreak havoc among numerically superior opponents with relative ease (pilot training and experience was CRUCIAL), 2) stacking penalty reduces stats of aircraft, so individually they are somehow less effective even though that shouldn't really be the case.